# Constructing Disjoint Steiner Trees in Sierpiński Graphs††thanks: Supported by the National Natural Science Foundation of China (Nos. 12201375, 12471329 and 12061059) and the Qinghai Key Laboratory of Internet of Things Project (2017-ZJ-Y21).
**Authors**:
- Chenxu Yang (School of Computer)
- Qinghai Normal University
- Xining
- Qinghai
- Ping Li (School of Mathematics and Statistics)
- Shaanxi Normal University
- Xi’an
- Shaanxi
- Yaping Mao
- Xining
- Qinghai
- Eddie Cheng (Department of Mathematics)
- Oakland University
- Rochester
- MI USA
- Ralf Klasing
- Université de Bordeaux
- Bordeaux INP
- CNRS
- LaBRI
- UMR
- Talence
> Corresponding author.
25 2 10.46298/fi.12478
Abstract
Let $G$ be a graph and $S⊂eq V(G)$ with $|S|≥ 2$ . Then the trees $T_{1},T_{2},·s,T_{\ell}$ in $G$ connecting $S$ are internally disjoint Steiner trees (or $S$ -Steiner trees) if $E(T_{i})\cap E(T_{j})=\emptyset$ and $V(T_{i})\cap V(T_{j})=S$ for every pair of distinct integers $1≤ i,j≤\ell$ . Similarly, if we only have the condition $E(T_{i})\cap E(T_{j})=\emptyset$ but without the condition $V(T_{i})\cap V(T_{j})=S$ , then they are edge-disjoint Steiner trees $S$ -Steiner trees. The generalized $k$ -connectivity, denoted by $\kappa_{k}(G)$ , of a graph $G$ , is defined as $\kappa_{k}(G)=\min\{\kappa_{G}(S)|S⊂eq V(G)\ \textrm{and}\ |S|=k\}$ , where $\kappa_{G}(S)$ is the maximum number of internally disjoint $S$ -Steiner trees. The generalized $k$ -edge-connectivity $\lambda_{k}(G)$ of $G$ is defined as $\lambda_{k}(G)=\min\{\lambda_{G}(S)\,|\,S⊂eq V(G)\ and\ |S|=k\}$ , where $\lambda_{G}(S)$ is the maximum number of edge-disjoint Steiner trees connecting $S$ in $G$ . These concepts are generalizations of the concepts of connectivity and edge-connectivity, and they can be used as measures of vulnerability of networks. It is, in general, difficult to compute these generalized connectivities. However, there are precise results for some special classes of graphs. In this paper, we obtain the exact value of $\lambda_{k}(S(n,\ell))$ for $3≤ k≤\ell^{n}$ , and the exact value of $\kappa_{k}(S(n,\ell))$ for $3≤ k≤\ell$ , where $S(n,\ell)$ is the Sierpiński graphs with order $\ell^{n}$ . As a direct consequence, these graphs provide additional interesting examples when $\lambda_{k}(S(n,\ell))=\kappa_{k}(S(n,\ell))$ . We also study the some network properties of Sierpiński graphs. Keywords: Steiner Tree; Generalized Connectivity; Sierpiński Graph. AMS subject classification 2010: 05C40, 05C85. volume: 194 issue: 1
Constructing disjoint Steiner trees in Sierpiński graphs
1 Introduction
All graphs considered in this paper are undirected, finite and simple. We refer the readers to [1] for graph theoretical notation and terminology not described here. For a graph $G$ , let $V(G)$ and $E(G)$ denote the set of vertices and the set of edges of $G$ , respectively. The neighborhood set of a vertex $v∈ V(G)$ is $N_{G}(v)=\{u∈ V(G)\,|\,uv∈ E(G)\}$ . The degree of a vertex $v$ in $G$ is denoted by $d(v)=|N_{G}(v)|$ . Denote by $\delta(G)$ ( $\Delta(G)$ ) the minimum degree (maximum degree) of the graph $G$ . For a vertex subset $S⊂eq V(G)$ , the subgraph induced by $S$ in $G$ is denoted by $G[S]$ and similarly $G[V\setminus S]$ for $G\setminus S$ or $G-S$ . Especially, $G-v$ is $G[V\setminus\{v\}]$ . Let $\overline{G}$ be the complement of $G$ . For a partition $\mathcal{P}=\{V_{1},V_{2},...,V_{t}\}$ of $V(G)$ , let $G/\mathcal{P}$ be the graph obtained from $G$ by deleting $\bigcup_{i∈[t]}E(G[V_{i}])$ and then identifying each $V_{i}$ , respectively. For any positive integers $n$ , we always use the convenient notation $[n]$ to denote the set $\{1,2,·s,n\}$ .
1.1 Generalized (edge-)connectivity
Connectivity and edge-connectivity are two of the most basic concepts of graph-theoretic measures. Such concepts can be generalized, see, for example, [16]. For a graph $G=(V,E)$ and a set $S⊂eq V(G)$ of at least two vertices, an $S$ -Steiner tree or a Steiner tree connecting $S$ (or simply, an $S$ -tree) is a subgraph $T=(V^{\prime},E^{\prime})$ of $G$ that is a tree with $S⊂eq V^{\prime}$ . Note that when $|S|=2$ a minimal $S$ -Steiner tree is just a path connecting the two vertices of $S$ .
Let $G$ be a graph and $S⊂eq V(G)$ with $|S|≥ 2$ . Then the trees $T_{1},T_{2},·s,T_{\ell}$ in $G$ are internally disjoint $S$ -trees if $E(T_{i})\cap E(T_{j})=\emptyset$ and $V(T_{i})\cap V(T_{j})=S$ for every pair of distinct integers $i,j$ , $1≤ i,j≤\ell$ . Similarly, if we only have the condition $E(T_{i})\cap E(T_{j})=\emptyset$ but without the condition $V(T_{i})\cap V(T_{j})=S$ , then they are edge-disjoint $S$ -trees (Note that while we do not have the condition $V(T_{i})\cap V(T_{j})=S$ , it is still true that $S⊂eq V(T_{i})\cap V(T_{j})$ as $T_{i}$ and $T_{j}$ are $S$ -trees.) The generalized $k$ -connectivity, denoted by $\kappa_{k}(G)$ , of a graph $G$ , is defined as $\kappa_{k}(G)=\min\{\kappa_{G}(S)|S⊂eq V(G)\ \textrm{and}\ |S|=k\}$ , where $\kappa_{G}(S)$ is the maximum number of internally disjoint $S$ -trees. The generalized local edge-connectivity $\lambda_{G}(S)$ is the maximum number of edge-disjoint $S$ -trees in $G$ . The generalized $k$ -edge-connectivity $\lambda_{k}(G)$ of $G$ is defined as $\lambda_{k}(G)=\min\{\lambda_{G}(S)\,|\,S⊂eq V(G)\ and\ |S|=k\}$ . Since internally disjoint $S$ -trees are edge-disjoint but not vice versa, it follows from the definitions that $\kappa_{k}(G)≤\lambda_{k}(G)$ . There are many results on generalized (edge-)connectivity; see the book [15] by Li and Mao.
For a graph $G$ and two distinct vertices $x,y$ of $G$ , the local connectivity $p_{G}(x,y)$ of $x$ and $y$ is defined as the maximum number of pairwise internally disjoint paths between $x$ and $y$ , and the local edge-connectivity $\lambda_{G}(x,y)$ is defined as the maximum number of pairwise edge-disjoint paths between $x$ and $y$ . The connectivity of $G$ is defined as $\kappa(G)=\min\{p_{G}(x,y)\,|\,x,y∈ V(G),\,x≠ y\}$ , and the edge-connectivity of $G$ is defined as $\lambda(G)=\min\{\lambda_{G}(x,y)\,|\,x,y∈ V(G),\,x≠ y\}$ . It is clear that when $|S|=2$ , $\lambda_{2}(G)$ is just the standard edge-connectivity $\lambda(G)$ of $G$ , $\kappa_{2}(G)=\kappa(G)$ , that is, the standard connectivity of $G$ . Thus $\kappa_{k}(G)$ and $\lambda_{k}(G)$ are the generalized connectivity of $G$ and the generalized edge-connectivity of $G$ , respectively.
As it is well-known, for any graph $G$ , we have polynomial-time algorithms to find the classical connectivity $\kappa(G)$ and edge-connectivity $\lambda(G)$ . Given two fixed positive integers $k$ and $\ell$ , for any graph $G$ the problem of deciding whether $\lambda_{k}(G)≥\ell$ can be solved by a polynomial-time algorithm. If $k\ (k≥ 3)$ is a fixed integer and $\ell\ (\ell≥ 2)$ is an arbitrary positive integer, the problem of deciding whether $\kappa(S)≥\ell$ is $NP$ -complete. For any fixed integer $\ell≥ 3$ , given a graph $G$ and a subset $S$ of $V(G)$ , deciding whether there are $\ell$ internally disjoint Steiner trees connecting $S$ , namely deciding whether $\kappa(S)≥\ell$ , is $NP$ -complete. For more details on the computational complexity of generalized connectivity and generalized edge-connectivity, we refer to the book [15].
In addition to being a natural combinatorial measure, generalized $k$ -connectivity can be motivated by its interesting interpretation in practice. For example, suppose that $G$ represents a network. If one wants to “connect” a pair of vertices of $G$ “minimally”, then a path is used to “connect” them. More generally, if one wants to “connect” a set $S$ of vertices of $G$ , with $|S|≥ 3$ , “minimally”, then it is desirable to use a tree to “connect” them. Such trees are precisely $S$ -trees, which are also used in computer communication networks (see [8]) and optical wireless communication networks (see [6]).
From a theoretical perspective, generalized edge-connectivity is related to Nash-Williams-Tutte theorem and Menger theorem; see [15]. The generalized edge-connectivity has applications in $VLSI$ circuit design. In this application, a Steiner tree is needed to share an electronic signal by a set of terminal nodes. Another application, which is our primary focus, arises in the Internet Domain. Imagine that a given graph $G$ represents a network. We arbitrarily choose $k$ vertices as nodes. Suppose one of the nodes in $G$ is a broadcaster, and all other nodes are either users or routers (also called switches). The broadcaster wants to broadcast as many streams of movies as possible, so that the users have the maximum number of choices. Each stream of movie is broadcasted via a tree connecting all the users and the broadcaster. So, in essence we need to find the maximum number Steiner trees connecting all the users and the broadcaster, namely, we want to get $\lambda(S)$ , where $S$ is the set of the $k$ nodes. Clearly, it is a Steiner tree packing problem. Furthermore, if we want to know whether for any $k$ nodes the network $G$ has above properties, then we need to compute $\lambda_{k}(G)=\min\{\lambda(S)\}$ in order to prescribe the reliability and the security of the network. For more details, we refer to the book [15].
1.2 Sierpiński graphs
In 1997, Klavžar and Milutinović introduced the concept of Sierpiński graph $S(n,\ell)$ in [11]. We denote $n$ -tuples $V^{n}$ by the set
$$
V^{n}=\{\langle u_{0}u_{1}\cdots u_{n-1}\rangle|\ u_{i}\in\{0,1,\ldots,\ell-1%
\}\textrm{ and }i\in\{0,1,\ldots,n-1\}\}.
$$
A word $u$ of size $n$ are denoted by $\langle u_{0}u_{1},·s,u_{n-1}\rangle$ in which $u_{i}∈\{0,...,\ell-1\}$ . The concatenation of two words $u=\langle u_{0}u_{1}·s u_{n-1}\rangle$ and $v=\langle v_{0}v_{1}·s v_{n-1}\rangle$ is denoted by $uv$ .
**Definition 1**
*The Sierpiński graph $S(n,\ell)$ is defined as below, for $n≥ 1$ and $\ell≥ 3$ , the vertex set of $S(n,\ell)$ consists of all $n$ -tuples of integers $0,1,·s,\ell-1$ . That is, $V(S(n,\ell))=V^{n}$ , where $uv=(u_{0}u_{1}·s u_{n-1},v_{0}v_{1}·s v_{n-1})$ is an edge of $E(S(n,\ell))$ if and only if there exists $d∈\{0,1,...,\ell-1\}$ such that: $(1)$ $u_{j}=v_{j}$ , if $j<d$ ; $(2)$ $u_{d}≠ v_{d}$ ; $(3)$ $u_{j}=v_{d}$ and $v_{j}=u_{d}$ , if $j>d$ .*
<details>
<summary>x1.png Details</summary>
### Visual Description
## Diagram: Interconnected Graph with Numerical Labels
### Overview
The image presents a complex graph consisting of interconnected nodes and edges. The nodes are labeled with two-digit numbers. The graph exhibits a symmetrical structure with triangular sub-components.
### Components/Axes
* **Nodes:** Represented by circles, each labeled with a two-digit number.
* **Edges:** Represented by lines connecting the nodes.
* **Labels:** Two-digit numbers assigned to each node.
### Detailed Analysis
The graph can be broken down into the following components:
* **Central Structure:** A vertical chain of nodes labeled 00, 01, 10, and 11.
* **Top Triangle:** Nodes 11, 12, and 13 form a triangle.
* **Left Triangle:** Nodes 00, 02, and 20 form a triangle.
* **Right Triangle:** Nodes 00, 03, and 30 form a triangle.
* **Bottom-Left Triangle:** Nodes 20, 21, 22, and 23 form a structure where 22 is connected to 20 and 23, and 21 is connected to 20 and 22, and 23 is connected to 21 and 22.
* **Bottom-Right Triangle:** Nodes 30, 31, 32, and 33 form a structure where 33 is connected to 30 and 32, and 31 is connected to 30 and 33, and 32 is connected to 31 and 33.
* **Outer Connections:** Node 12 is connected to 21, and node 13 is connected to 31, and node 23 is connected to 32.
Specific node connections:
* 00 is connected to 01, 02, and 03.
* 01 is connected to 00 and 10.
* 10 is connected to 01 and 11.
* 11 is connected to 10, 12, and 13.
* 12 is connected to 11, 13, and 21.
* 13 is connected to 11, 12, and 31.
* 20 is connected to 02, 21, 22, and 23.
* 30 is connected to 03, 31, 32, and 33.
* 21 is connected to 12, 20, 22, and 23.
* 31 is connected to 13, 30, 32, and 33.
* 22 is connected to 20, 21, and 23.
* 33 is connected to 30, 31, and 32.
* 23 is connected to 20, 21, and 22.
* 32 is connected to 30, 31, and 33.
* 02 is connected to 00 and 20.
* 03 is connected to 00 and 30.
### Key Observations
* The graph exhibits symmetry around the central vertical chain of nodes.
* The two-digit labels appear to follow a pattern, but the logic is not immediately clear.
* The graph contains multiple triangular sub-structures.
### Interpretation
The diagram likely represents a network or system where the nodes represent entities and the edges represent relationships between them. The numerical labels could represent identifiers, attributes, or states of the entities. The triangular structures might indicate clusters or groups of related entities. Without further context, it's difficult to determine the specific meaning of the graph, but the structure suggests a complex and interconnected system.
</details>
Figure 1: $S(2,4)$
Sierpiński graph $S(2,4)$ is shown in Figure 1. Note that $S(n,\ell)$ can be constructed recursively as follows: $S(1,\ell)$ is isomorphic to $K_{\ell}$ , which vertex set is 1-tuples set $\{0,...,\ell-1\}$ . To construct $S(n,\ell)$ for $n>1$ , we take copies of $\ell$ times $S(n-1,\ell)$ and add the letter $i$ on the top of the vertices in $i$ -th copy of $S(n-1,\ell)$ , denoted by $S^{i}(n,\ell)$ . Note that there is exactly one edge (bridge edge) between $S^{i}(n,\ell)$ and $S^{j}(n,\ell)$ , $i≠ j$ , namely the edge between vertices $\langle ij·s j\rangle$ and $\langle ji·s i\rangle$ .
The vertices $(\underbrace{i,i,·s,i}_{n})$ , $i∈\{0,1,...,\ell-1\}$ are the extreme vertices of $S(n,\ell)$ . Note that an extreme vertex $u$ of $S(n,\ell)$ has degree $d(u)=\ell-1$ . For $i∈\{0,...,\ell-1\}$ and $n≥ 2$ , let $S^{i}(n-1,\ell)$ denote the subgraph of $S(n,\ell)$ induced by the vertices of the form $\{\langle iu_{1}·s u_{n-1}\rangle\,|\,0≤ u_{i}≤\ell-1\}$ . The vertex set $V(S(n,\ell))$ can be partitioned into $\ell$ parts $V(S^{0}(n-1,\ell)),V(S^{1}(n-1,\ell)),...,V(S^{\ell-1}(n-1,\ell))$ . For each $0≤ i≤\ell-1$ , $S^{i}(n-1,\ell)$ is isomorphic to $S(n-1,\ell)$ . Note that $V(S(n,\ell))=V(S^{0}(n-1,\ell))\cup·s\cup V(S^{\ell-1}(n-1,\ell))$ and $S(n,\ell)$ is the graph obtained from $S^{0}(n-1,\ell),...,S^{\ell-1}(n-1,\ell)$ by adding exactly one edge (bridge edge) between $S^{i}(n-1,\ell)$ and $S^{j}(n-1,\ell)$ , $i≠ j$ , and the bridge edge joins $\langle ij·s j\rangle$ and $\langle ji·s i\rangle$ (notices that $\langle ij·s j\rangle$ and $\langle ji·s i\rangle$ are extreme vertices of $S^{i}(n-1,\ell)$ and $S^{j}(n-1,\ell)$ , respectively, if we regard $S^{i}(n-1,\ell)$ and $S^{j}(n-1,\ell)$ as two copies of $S(n-1,\ell)$ ).
Sierpiński graphs generalize Hanoi graphs which can be viewed as “discrete” finite versions of a Sierpiński gasket [23, 10]. Xue considered the Hamiltionicity and path $t$ -coloring of Sierpiński-like graphs in [25]; furthermore, they proved that $Val(S(n,k))=Val(S[n,k])=\lfloor k/2\rfloor$ , where $Val(S(n,k))$ is the linear arboricity of Sierpiński graphs. We remark that although Sierpiński graphs are not regular, they are “almost” regular as the extreme vertices have degrees one less than the degrees of non-extreme vertices.
1.3 Preliminaries and our results
Chartrand et al. [4] and Li et al. [16] obtained the exact value of $\kappa_{k}(K_{n})$ .
**Theorem 1.1 ([16,4])**
*For every two integers $n$ and $k$ with $2≤ k≤ n$ ,
$$
\kappa_{k}(K_{n})=\lambda_{k}(K_{n})=n-\lceil k/2\rceil.
$$*
The following result is on the Hamiltonian decomposition of Sierpiński graphs.
**Theorem 1.2**
*[25] $(1)$ For even $\ell≥ 2$ , $S(n,\ell)$ can be decomposed into edge disjoint union of $\frac{\ell}{2}$ Hamiltonian paths of which the end vertices are extreme vertices. $(2)$ For odd $\ell≥ 3$ , there exist $\frac{\ell-1}{2}$ edge-disjoint Hamiltonian paths whose two end vertices are extreme vertices in $S(n,\ell)$ .*
In fact, Theorem 1.2 is used in the proof of Theorem 1.5, which give an lower bound for the generalized $k$ -edge connectivity of Sierpiński graphs $S(n,\ell)$ . We require the following result.
**Theorem 1.3 ([5,26])**
*Suppose that $G$ is a complete graph with $V(G)=\{v_{0},·s,v_{N-1}\}$ . If $N=2n$ , then $G$ can be decomposed into $n$ Hamiltonian paths
$$
\{i\in\{1,2,\ldots,n\}:\ L_{i}=v_{0+i}v_{1+i}v_{2n-1+i}v_{2+i}v_{2n-2+i}\cdots
v%
_{n+1+i}v_{n+1}\},
$$
where the subscripts take modulo $2n$ . If $N=2n+1$ , then $G$ can be decomposed into $n$ Hamiltonian paths
$$
\{i\in\{1,2,\ldots,n\}:\ L_{i}=\ v_{0+i}v_{1+i}v_{2n+i}v_{2+i}v_{2n-1+i}\cdots
v%
_{n+i}v_{n+1+i}\}
$$
and a matching $M=\{v_{n-i}v_{n+1}:\ i∈\{1,2,...,n\}\}$ , where the subscripts take modulo $2n+1$ .*
The following result is derived from Theorem 1.3, and we will use it later.
**Corollary 1.4**
*Let $s$ be an integer with $s≤\frac{N}{2}$ . Suppose that $G$ is the complete graph with $V(G)=\{v_{1},·s,v_{N}\}$ and $\mathcal{S}=\{\ \{v_{i_{1}},v_{i_{2}}\}:i∈\{1,2,...,s\}\}$ is a collection of pairwise disjoint $2$ -subsets of $V(G)$ . Then there are $s$ edge-disjoint Hamiltonian paths $L_{1},·s,L_{s}$ such that $v_{i_{1}},v_{i_{2}}$ are endpoints of $L_{i}$ .*
Our main result is as follows.
**Theorem 1.5**
*$(i)$ For $3≤ k≤\ell$ , we have
$$
\kappa_{k}(S(n,\ell))=\lambda_{k}(S(n,\ell))=\ell-\left\lceil k/2\right\rceil.
$$ $(ii)$ For $\ell+1≤ k≤{\ell}^{n}$ , we have
$$
\kappa_{k}(S(n,\ell))\leq\left\lfloor\ell/2\right\rfloor\mbox{ and }\lambda_{k%
}(S(n,\ell))=\left\lfloor\ell/2\right\rfloor.
$$*
2 Proof of Theorem 1.5
This section is organized as follows. We first introduce some notations, operations and auxiliary graphs that are used in the proof. Then we give a transitional theorem (Theorem 2.1), which is the main tool for the proof of Theorem 1.5. Theorem 2.1 will be proved by constructing an algorithm, and the proof takes up almost entire Section 2 (the proof is divided into two parts: the construction of algorithm and the proof of its correctness). Finally, we complete the proof of Theorem 1.5 by using Theorem 2.1.
2.1 Notations, operations and auxiliary graphs
For $1≤ s<n$ , by the definition of the Sierpiński graph $S(n,\ell)$ , we have that $G=S(n,\ell)$ consists of $\ell^{n-s}$ Sierpiński graphs $S(s,\ell)$ , and we call each such $S(s,\ell)$ an $s$ -atom of $G$ . Note that $\mathcal{P}=\{V(H):\ H\mbox{ is an }s\mbox{-atom of }G\}$ is a partition of $V(G)$ . We use $G_{s}$ to denote the graph $G/\mathcal{P}$ . It is obvious that $G_{s}$ is a simple graph and $G_{s}=S(n-s,\ell)$ . Note that $G_{n}$ is a single vertex and $G_{0}=G$ . For a vertex $u$ of $G_{s}$ , we use $A_{s,u}$ to denote the $s$ -atom of $G$ which is contracted into $u$ . See Figures 4, 4 and 4 for illustrations.
| Notation | Corresponding interpretation |
| --- | --- |
| $s$ -atom | A copy of $S(s,\ell)$ in $G=S(n,\ell)$ , where $0≤ s≤ n$ . |
| $G_{s}$ | The graph obtained from $G$ by contracting all $\ell^{n-s}$ $s$ -atoms in $G$ . |
| $A_{u,s}$ | The $s$ -atom in $G$ which is contracted into $u$ , where $u∈ V(G_{s})$ . |
| $H^{u}$ | $G_{s-1}$ can be obtained form $G_{s}$ by expending each $u∈ V(G_{s})$ to a complete graph $H^{u}=K_{\ell}$ . |
| $u^{e},v^{e}$ | If $e=uv$ is an edge of $G_{s}$ , then $e$ is also an edge of $G_{s-1}$ . let $u_{e},v_{e}$ denote endpoints of $e$ in $G_{s-1}$ such that $u_{e}∈ H^{u}$ and $v_{e}∈ H^{v}$ . |
| $U$ | The set of labelled vertices in $G$ ( $|U|=k$ ). |
| $U^{s}$ | The set of labelled vertices in $G_{s}$ . |
| $W^{u}$ | The set of labelled vertices in $H^{u}$ . |
Table 1: Notations for and their meanings.
Note that $G_{s-1}$ is the graph obtained from $G_{s}$ by replacing each vertex $u$ with a complete graph $K_{\ell}$ . We denote by $H^{u}$ the complete graph replacing $u$ (see Figures 4 and 4 for illustrations). For an edge $e=uv$ of $G_{s}$ , $e$ is also an edge of $G_{s-1}$ . We use $u_{e}$ and $v_{e}$ to denote the ends of $e$ in $G_{s-1}$ , respectively, such that $u_{e}∈ V(H^{u})$ and $v_{e}∈ V(H^{v})$ . (See Figure 5 for an illustration.)
Fix a subset $U$ of $V(G)$ arbitrary such that $|U|=k$ , where $3≤ k≤\ell$ . For the graph $G_{s}$ , a vertex $u$ of $G_{s}$ is labelled if $V(A_{s,u})\cap U≠\emptyset$ , and $u$ is unlabelled otherwise. We use $U^{s}$ to denote the set of labelled vertices of $G_{s}$ . For a labelled vertex $u$ of $G_{s}$ , let $W^{u}=V(H^{u})\cap U^{s-1}$ , that is, the set of labelled vertices in $H^{u}$ . See Figure 4 as an example, if $U=\{u_{000},u_{001},u_{013},u_{122}\}$ , then $U^{1}=\{u_{00},u_{01},u_{12}\}$ and $W^{u_{0}}=\{u_{00},u_{01}\}$ in Figure 4. For ease of reading, above notations and their meanings are summarized in Table 1.
We construct an out-branching $\overrightarrow{T}$ with vertex set $V(\overrightarrow{T})=\bigcup_{i∈\{0,1,...,n\}}\{x:\ x∈ V(G_{i})\}$ and arc set $A(\overrightarrow{T})=\{(x,y):\ y∈ V(H^{x})\}$ . The root of $\overrightarrow{T}$ is denoted by $v_{root}$ (it is clear that $V(G_{n})=\{v_{root}\}$ ). For two different vertices $x,y$ of $V(\overrightarrow{T})$ , if there is a direct path from $x$ to $y$ , then we say that $x\prec y$ , and denote the directed path by $x\overrightarrow{T}y$ . The following result is straightforward.
**Fact 1**
*If $x_{i}∈ V(\overrightarrow{T})$ for $i∈[p]$ and $x_{1}\prec x_{2}\prec...\prec x_{p}$ , then $\sum_{i∈[p]}|W^{x_{i}}|≤ k+p-1$ .*
$u_{000}$ $u_{001}$ $u_{002}$ $u_{003}$ $u_{010}$ $u_{012}$ $u_{011}$ $u_{013}$ $u_{020}$ $u_{021}$ $u_{030}$ $u_{031}$ $u_{022}$ $u_{023}$ $u_{032}$ $u_{033}$ $u_{100}$ $u_{101}$ $u_{110}$ $u_{111}$ $u_{102}$ $u_{103}$ $u_{112}$ $u_{113}$ $u_{120}$ $u_{121}$ $u_{130}$ $u_{131}$ $u_{122}$ $u_{123}$ $u_{132}$ $u_{133}$ $u_{200}$ $u_{201}$ $u_{210}$ $u_{211}$ $u_{300}$ $u_{301}$ $u_{310}$ $u_{311}$ $u_{202}$ $u_{203}$ $u_{212}$ $u_{213}$ $u_{302}$ $u_{303}$ $u_{312}$ $u_{313}$ $u_{220}$ $u_{221}$ $u_{230}$ $u_{231}$ $u_{320}$ $u_{321}$ $u_{330}$ $u_{331}$ $u_{222}$ $u_{223}$ $u_{232}$ $u_{233}$ $u_{322}$ $u_{323}$ $u_{332}$ $u_{333}$ $A(2,u_{0})$ $A(2,u_{1})$
Figure 2: $G=G_{0}=S(3,4)$
$u_{00}$ $u_{01}$ $u_{02}$ $u_{03}$ $u_{10}$ $u_{11}$ $u_{12}$ $u_{13}$ $u_{20}$ $u_{21}$ $u_{30}$ $u_{31}$ $u_{22}$ $u_{23}$ $u_{32}$ $u_{33}$ $H^{u_{0}}$ $H^{u_{1}}$
Figure 3: The graph $G_{1}$
$u_{0}$ $u_{1}$ $u_{2}$ $u_{3}$
Figure 4: The graph $G_{2}$
2.2 A transitional theorem and its proof
In order to prove our main result (Theorem 1.5), we first prove the following transitional theorem by constructing $\ell-\left\lceil k/2\right\rceil$ internally disjoint $U$ -Steiner trees in $G$ .
**Theorem 2.1**
*Let $G=S(n,\ell)$ and $U⊂eq V(G)$ with $|U|=k$ ( $U$ is defined before), and let $c=\ell-\left\lceil k/2\right\rceil$ . For $0≤ s≤ n$ , $G_{s}$ has $c$ internally disjoint $U^{s}$ -Steiner trees $T_{1},·s,T_{c}$ such that the following statement holds.
1. For each $u∈ U^{s}$ (say $u∈ W^{v}$ and hence $v$ is a labelled vertex of $G_{s+1}$ if $s≠ n$ ) and $i∈[c]$ , $d_{T_{i}}(u)≤ 2$ .*
The rest part of this subsection is the proof of Theorem 2.1.
Proof of Theorem 2.1 Suppose that $s^{\prime}$ is the minimum integer such that $G_{s^{\prime}}$ has exactly one labelled vertex (note that $s^{\prime}$ exists since $G_{n}$ consists of a single labelled vertex). This means that $G_{s^{\prime}-1}$ has at least two labelled vertices. Let $v$ be the labelled vertex in $G_{s^{\prime}}$ . Then $U⊂eq A_{s^{\prime},v}$ . Thus, in order to find $c$ internally disjoint $U$ -Steiner trees of $G$ satisfying ( $\star$ ), we only need to find $c$ internally disjoint $U$ -Steiner trees in $A_{s^{\prime},v}$ satisfying ( $\star$ ). Hence, without loss of generality, we can assume that $G$ is a graph with
$$
|W^{v_{root}}|=|U^{n-1}|\geq 2. \tag{1}
$$
Therefore, $|U^{i}|≥ 2$ for each $i≤ n-1$ . $u_{11}$ $u_{12}$ $u_{13}$ $u_{24}$ $u_{21}$ $u_{22}$ $u_{23}$ $u_{24}$ $u_{31}$ $u_{32}$ $u_{41}$ $u_{42}$ $u_{33}$ $u_{34}$ $u_{43}$ $u_{44}$ $e$ $u_{e}$ $v_{e}$ $u$ $v$ $e$ $G_{1}$ $G_{2}$
Figure 5: From graph $G_{1}$ to $G_{2}$ by $u_{e}v_{e}$ contraction edge $uv$ .
The proof is technical and is via induction. Note that since $G_{n}$ is a single vertex and $G=G_{0}$ , the induction is from $G_{n}$ to $G_{0}$ . The basic idea is to use the $U^{s}$ -Steiner trees of $G_{s}$ to construct appropriate $U^{s-1}$ -Steiner trees of $G_{s-1}$ . Since each vertex in $G_{s}$ corresponds to a complete graph in $G_{s-1}$ , it is not a straightforward process to extend $U^{s}$ -Steiner trees of $G_{s}$ to $U^{s-1}$ -Steiner trees of $G_{s-1}$ .
If $s=n$ , then each $T_{i}$ in $G_{n}$ is the empty graph and the result holds. Thus, suppose $s≤ n-1$ . Hence, the labelled vertex $v$ of $G_{s+1}$ always exists. The following implies that the result holds for $s=n-1$ . Note that $U^{n-1}=W^{v_{root}}$ .
**Claim 1**
*If $s=n-1$ , then we can construct $c$ internally disjoint $U^{s}$ -Steiner trees, say $T_{1},...,T_{c}$ , such that for each $i∈[c]$ and $u∈ U^{s}$ , $d_{T_{i}}(u)≤ 2$ . Moreover,
1. $(i)$ If $|U^{s}|≤\lceil k/2\rceil$ , then for each $i∈[c]$ and $u∈ U^{s}$ , $d_{T_{i}}(u)=1$ ;
1. $(ii)$ otherwise, for each $u∈ U^{s}$ , there are at most $|U^{s}|-\lceil k/2\rceil$ internally disjoint $U^{s}$ -Steiner trees $T_{i}$ such that $d_{T_{i}}(u)=2$ .*
* Proof*
Note that $G_{n-1}=S(1,\ell)=K_{\ell}$ . Suppose $|U^{n-1}|≤\left\lceil k/2\right\rceil$ . Then $|V(G_{n-1})-U^{n-1}|≥\ell-\left\lceil k/2\right\rceil=c$ and we can choose $c$ stars of $\{x\vee U^{n-1}:\ x∈ V(G_{n-1})-U^{n-1}\}$ as internally disjoint $U^{n-1}$ -Steiner trees $T_{1},...,T_{c}$ . It is easy to verify that $d_{T_{i}}(u)=1$ for $i∈[c]$ and $u∈ U^{n-1}$ . Suppose $|U^{n-1}|≥\lceil k/2\rceil$ . Since $|U^{n-1}|≤ k$ , it follows that $0≤|U^{n-1}|-\left\lceil k/2\right\rceil≤\lfloor|U^{n-1}|/2\rfloor$ . By Corollary 1.4, there are $|U^{n-1}|-\left\lceil k/2\right\rceil$ edge-disjoint Hamiltonian paths in $G_{n-1}[U^{n-1}]$ . So, we can choose $c$ internally disjoint $U^{n-1}$ -Steiner trees consisting of $|U^{n-1}|-\left\lceil k/2\right\rceil$ edge-disjoint Hamiltonian paths in $G_{n-1}[U^{n-1}]$ and $\ell-|U^{n-1}|$ stars $\{x\vee U^{n-1}:\ x∈ V(G_{n-1})-U^{n-1}\}$ . It is easy to verify that $d_{T_{i}}(u)≤ 2$ for $i∈[c]$ and $u∈ U^{n-1}$ , and there are at most $|U^{n-1}|-\left\lceil k/2\right\rceil$ internally disjoint $U^{n-1}$ -Steiner trees $T_{i}$ such that $d_{T_{i}}(u)=2$ . ∎
Our proof is a recursive process that the $c$ internally disjoint $U^{s}$ -Steiner trees of $G_{s}$ are constructed by using the $c$ internally disjoint $U^{s+1}$ -Steiner trees of $G_{s+1}$ . We will find some ways to construct the $c$ internally disjoint $U^{s}$ -Steiner trees such that $(\star)$ holds. In the finial step, the $c$ internally disjoint $U$ -Steiner trees in $G_{0}=G$ will be obtained. We have proved that the result holds for $s∈\{n,n-1\}$ . Now, suppose that we have constructed $c$ internally disjoint $U^{s+1}$ -Steiner trees, say $\mathcal{F}=\{F_{1},...,F_{c}\}$ , and the trees satisfy ( $\star$ ), where $s∈\{0,...,n-1\}$ . We need to construct $c$ internally disjoint $U^{s}$ -Steiner trees $\mathcal{T}=\{T_{1},...,T_{c}\}$ of $G_{s}$ satisfying ( $\star$ ).
Recall that each edge of $G_{s+1}$ is also an edge of $G_{s}$ . In addition, let $E_{u,i}$ denote the set of edges in $F_{i}$ incident with the labelled vertex $u$ in $G_{s}$ and let $V_{u,i}=\{u_{e}:\ e∈ E_{u,i}\}$ (recall that the edge $e=uw$ in $G_{s+1}$ is denoted by $e=u_{e}w_{e}$ in $G_{s}$ , where $u_{e}∈ V(H^{u})$ and $w_{e}∈ V(H^{w})$ ). Note that $E_{u,i}$ and $V_{u,i}$ are the subsets of $E(G_{s})$ and $V(G_{s})$ , respectively.
Our aim is to construct internally disjoint $U^{s}$ -Steiner trees $T_{1},...,T_{c}$ that is obtained from $U^{s+1}$ -Steiner trees $F_{1},...,F_{c}$ . So, we need to “blow” up each vertex $w$ of $G_{s+1}$ into $H^{w}$ and find internally disjoint $W^{w}$ -Steiner trees $F_{1}^{w},...,F_{c}^{w}$ of $H^{w}$ ( $F_{i}^{w}$ may be the empty graph if $w$ is a unlabelled vertex) such that
$$
\mathcal{T}=\left\{T_{i}=F_{i}\cup\bigcup_{w\in V(G_{s+1})}F_{i}^{w}:\ i\in[c]%
\right\}.
$$
Each $F_{i}^{w}$ can be constructed as follows.
- If $w$ is a labelled vertex of $G_{s+1}$ , then $w$ is in each $U^{s+1}$ -Steiner tree of $\mathcal{F}$ . We choose $c$ internally disjoint $W^{w}$ -Steiner trees $F_{1}^{w},...,F_{c}^{w}$ of $H^{w}$ such that $V_{w,i}⊂eq V(F_{i}^{w})$ for each $i∈[c]$ .
- If $w$ is an unlabelled vertex, then $w$ is in at most one $U^{s+1}$ -Steiner tree of $\mathcal{F}$ . For each $i∈[c]$ , if $w∈ V(F_{i})$ , then let $F_{i}^{w}$ be a spanning tree of $H^{w}$ ; if $w∉ V(F_{i})$ , then let $F_{i}^{w}$ be the empty graph.
For $i∈[c]$ , let $E_{i}=E(F_{i})\cup\bigcup_{w∈ V(G_{s+1})}E(F_{i}^{w})$ and let $T_{i}=G_{s}[E_{i}]$ . It is obvious that $T_{1},...,T_{c}$ are internally disjoint $U^{s}$ -Steiner trees. We need to ensure that each labelled vertex of $G^{s}$ satisfies ( $\star$ ).
In fact, without loss of generality, we only need to choose an arbitrary labelled vertex $u∈ G_{s+1}$ and prove that each vertex of $W^{u}$ satisfies ( $\star$ ). This is because $F_{i}^{w}$ is clear when $w$ is an unlabelled vertex (in the case $F_{i}^{w}$ is either a spanning tree of $H^{w}$ or the empty graph). By Eq. (1) and Fact 1, $|W^{u}|≤ k-1$ since $u≠ v_{root}$ .
Since $d_{F_{i}}(u)≤ 2$ for each $i∈[c]$ and $u∈ U^{s}$ , it follows that $|E_{u,i}|=|V_{u,i}|≤ 2$ . Therefore, we can divide $F_{1},...,F_{c}$ into the following five types on $u$ :
Type 1: $|V_{u,i}|=1$ and $V_{u,i}⊂eq W^{u}$ .
Type 2: $|V_{u,i}|=1$ and $V_{u,i}\nsubseteq W^{u}$ .
Type 3: $|V_{u,i}|=2$ and $V_{u,i}⊂eq W^{u}$ .
Type 4: $|V_{u,i}|=2$ and $V_{u,i}\cap W^{u}=\emptyset$ .
Type 5: $|V_{u,i}|=2$ and $|V_{u,i}\cap W^{u}|=1$ .
If $F_{i}$ is a graph of Type $j\ (1≤ j≤ 5)$ on $u$ , then we also call $F_{i}^{u}$ a $W^{u}$ -Steiner tree of Type $j$ . Suppose there are $n_{j}(u)$ trees $F_{i}$ that belong to Type $j$ on $u$ for $j∈[5]$ . Then
$$
\displaystyle\sum_{i=1}^{5}n_{j}(u)=c. \tag{2}
$$
It is obvious that $n_{j}(v_{root})=0$ for each $j∈[5]$ . Let
$$
R(u)=V(H^{u})-W^{u}-\bigcup_{i\in[c]}V_{u,i}.
$$
Note that $R(u)$ is the set of unlabelled vertices in $G_{s}$ , which will be used to construct the $W^{u}$ -Steiner trees in $H^{u}$ . It is clear that
$$
|R(u)|=\ell-|W^{u}|-[n_{2}(u)+2n_{4}(u)+n_{5}(u)]. \tag{3}
$$
<details>
<summary>x2.png Details</summary>
### Visual Description
## Diagram: Graphical Representation of Topological Structures
### Overview
The image presents a series of diagrams illustrating different topological structures, categorized by "Type" and "Method." Each diagram consists of an oval shape, labeled components, and connecting lines, representing relationships between these components. The diagrams are grouped into distinct methods, each enclosed within a dotted-line rectangle.
### Components/Axes
* **Oval Shape:** Represents a base structure, with a curved line at the bottom.
* **Wu:** Label positioned inside the top half of the oval.
* **Hu:** Label positioned outside the bottom half of the oval.
* **Type Labels:** "Type 1," "Type 2," "Type 3," "Type 4," "Type 5" are located below each diagram in the top row and within the dotted rectangles in the subsequent rows.
* **Method Labels:** "Method 1," "Method 1a," "Method 2," "Method 3" are located above the diagrams within their respective dotted rectangles.
* **Points:** Small black circles representing nodes or points within the structure.
* **Lines:** Solid and dotted lines connecting the points and components.
* **x, y:** Labels indicating specific points in some diagrams.
### Detailed Analysis
**Top Row: Basic Types**
* **Type 1:** An oval with a horizontal line inside, connecting to a point on the top of the oval.
* **Type 2:** An oval with a curved line inside.
* **Type 3:** An oval with a curved line inside, connecting to two points on the top of the oval.
* **Type 4:** An oval with a curved line inside, and a line intersecting the oval.
* **Type 5:** An oval with a curved line inside, connecting to a point on the top of the oval.
**Method 1**
* Enclosed in a dotted rectangle.
* **Type 1:** An oval with a curved line inside. Several lines connect a point labeled "x" outside the oval to points inside the oval. A dotted line connects two points inside the oval.
* **Type 3:** Similar to Type 1, but with a different arrangement of connecting lines.
**Method 1a**
* Enclosed in a dotted rectangle.
* **Type 1:** An oval with a curved line inside. Several curved lines connect points inside the oval. A dotted line connects two points inside the oval.
* **Type 3:** Similar to Type 1, but with a different arrangement of curved lines.
**Method 2**
* Enclosed in a dotted rectangle.
* **Type 2:** An oval with a curved line inside. Several lines connect a point labeled "x" outside the oval to points inside the oval. A dotted line connects two points inside the oval.
* **Type 5:** Similar to Type 2, but with a different arrangement of connecting lines.
**Method 3**
* Enclosed in a dotted rectangle.
* **Type 4:** An oval with a curved line inside. Several lines connect a point labeled "x" outside the oval to points inside the oval. A dotted line connects two points inside the oval. Another line intersects the oval at point "y".
### Key Observations
* The diagrams represent different configurations of topological structures.
* The "Type" labels seem to indicate variations in the basic structure.
* The "Method" labels suggest different ways of modifying or connecting these structures.
* Dotted lines appear to represent a specific type of connection or relationship.
* The points labeled "x" and "y" likely represent specific nodes or points of interaction.
### Interpretation
The image likely illustrates different types of topological structures and methods for manipulating or connecting them. The diagrams could represent various mathematical or physical systems, where the ovals represent a base structure, and the lines represent relationships or interactions between components. The different "Types" and "Methods" likely represent different configurations or manipulations of these structures. The dotted lines might represent a specific type of interaction or connection, while the points labeled "x" and "y" could represent specific nodes or points of interest within the system. The image provides a visual representation of abstract concepts, potentially related to graph theory, network analysis, or other fields dealing with topological structures.
</details>
Figure 6: Five types and methods 1, 1a, 2 and 3.
We have the following five methods to choose $F_{1}^{u},...,F_{c}^{u}$ for the labelled vertex $u∈ G_{s+1}$ (see Figure 6).
Method 1 If $F_{i}$ is a graph of Type 1 and $|W^{u}|≥ 2$ , or $F_{i}$ is a graph of Type 3, then let $F^{u}_{i}=x\vee W^{u}$ , where $x∈ R(u)$ .
Method 1a If $F_{i}$ is a graph of Type 1 and $|W^{u}|≥ 2$ , then let $F^{u}_{i}$ be a Hamiltonian path of $H^{u}[W^{u}]$ such that the vertex in $V_{u,i}$ is an endpoint of this Hamiltonian path; if $F_{i}$ is a graph of Type 3, then let $F^{u}_{i}$ be a Hamiltonian path of $H^{u}[W^{u}]$ such that the endpoints of $F_{i}$ are two vertices in $V_{u,i}$ .
Method 2 If $F_{i}$ is a graph of Type 2 or Type 5, say $x$ is the only vertex of $V_{u,i}$ with $x∉ W^{u}$ , then let $F^{u}_{i}=x\vee W^{u}$ .
Method 3 If $F_{i}$ is a graph of Type 4, say $V_{u,i}=\{x,y\}$ , then let $F^{u}_{i}=xy\cup(x\vee W^{u})$ .
Method 4 If $F_{i}$ is a graph of Type 1 and $|W^{u}|=1$ , then let $F_{i}^{u}$ be the empty graph.
It is worth noting that the method is deterministic if $F_{i}^{u}$ is a graph of Types 2, 4 and 5, or $F_{i}^{u}$ is a graph of Type 1 and $|W^{u}|=1$ . So we firstly construct these $F_{i}^{u}$ s by using Methods 2, 3 and 4, respectively, and then construct $F^{u}_{i}$ s of Type 1 with $|W^{u}|≥ 2$ and construct $F^{u}_{i}$ s of Type 3 by using Method 1 or Method 1a.
The following is an algorithm constructing $c$ internally disjoint $U$ -Steiner trees of $G$ .
Input: $U$ , $c$ and $G=S(n,\ell)$ with $|W^{root}|≥ 2$
Output: $c$ internally disjoint $U$ -Steiner trees $T^{\prime}_{1},T^{\prime}_{2},...,T^{\prime}_{c}$ of $G$
1 $T^{\prime}_{1},T^{\prime}_{2},...,T^{\prime}_{c}$ are empty graphs;
2 $s=n$ //(* in initial step, $G^{n}$ is a single vertex*);
3 for $s≥ 1$ do
4 for each vertex $u$ of $G_{s}$ do
5 if $u$ is an unlabelled vertex then
6 for $1≤ i≤ c$ do
7 if $u$ is a vertex of $F_{i}$ then
8 $F_{i}^{u}$ is a spanning tree of $H^{u}$ ;
9 else
10 $\ F_{i}^{u}$ is is the empty graph;
11 end if
12 $T^{\prime}_{i}=T^{\prime}_{i}\cup F_{i}^{u}$ ;
13 $i=i+1$ ;
14
15 end for
16
17 end if
18
19 if $u$ is a labelled vertex then
20 $\mu=|R(u)|$ ;
21 for $1≤ i≤ c$ do
22
23 if $T^{\prime}_{i}$ is of Type 2 or Type 5 then
24 construct $F_{i}^{u}$ by using Method 2;
25
26 end if
27
28 if $T^{\prime}_{i}$ is of Type 4 then
29 construct $F_{i}^{u}$ by using Method 3;
30
31 end if
32
33 if $T^{\prime}_{i}$ is of Type 1 and $|W^{u}|=1$ then
34 construct $F_{i}^{u}$ by using Method 4;
35
36 end if
37
38 if either $T^{\prime}_{i}$ is of Type 1 and $|W^{u}|≥ 2$ , or $T^{\prime}_{i}$ is of Type 3 then
39 if $\mu>0$ then
40 construct $F_{i}^{u}$ by using Method 1;
41 $\mu=\mu-1$ ;
42
43 end if
44
45 if $\mu≤ 0$ then
46 construct $F_{i}^{u}$ by using Method 1a;
47
48 end if
49
50 end if
51
52 $T^{\prime}_{i}=T^{\prime}_{i}\cup F_{i}^{u}$ ;
53 $i=i+1$ ;
54
55 end for
56
57 end if
58
59 end for
60 $s=s-1$ ;
61
62 end for
Algorithm 1 The construction of $c$ internally disjoint $U$ -Steiner trees of $G$
Algorithm 1 is an algorithm for finding $c$ internally disjoint $U$ -Steiner trees of $G$ . For each step of $s$ , the algorithm will construct $c$ internally disjoint $U^{s-1}$ -steiner trees $T^{\prime}_{1},T^{\prime}_{2},...,T^{\prime}_{c}$ . For convenience, we denote each $T_{i}^{\prime}$ by $T_{i}^{s}$ after the $s$ step of outer “for”, that is, $T_{1}^{s},T_{2}^{s},...,T_{c}^{s}$ are $c$ internally disjoint $U^{s}$ -Steiner trees in $G^{s}$ generated in Algorithm 1. If Algorithm 1 is correct, then Lines 16–40 indicate that $d_{T^{\prime}_{i}}(x)≤ 2$ for any $x∈ W^{u}$ , and hence ( $\star$ ) always holds.
We now check the correctness of the algorithm. Since $F_{i}^{u}$ is deterministic when $u$ is an unlabelled vertex (Lines 5–15 of Algorithm 1), and $F_{i}^{u}$ is deterministic if $u$ is a labelled vertex and either $F_{i}^{u}$ is a graph of Types 2, 4 and 5, or $F_{i}^{u}$ is a graph of Type 1 and $|W^{u}|=1$ (Lines 19–27 of Algorithm 1), we only need to talk about the labelled vertex $u$ with $|W^{u}|≥ 2$ and check the correctness of Lines 28–36 of Algorithm 1. That is, to ensure that there exist $n_{1}(u)$ $F_{i}^{u}$ s of Type 1 and $n_{3}(u)$ $F_{i}^{u}$ s of Type 3. If $F_{i}^{u}$ is constructed by Method 1, then $F_{i}^{u}$ is a star with center in $R(u)$ (say the center of the star is $a_{i}$ ); if $F_{i}^{u}$ is constructed by Method 1a, then $F_{i}^{u}$ is a Hamiltonian path of $H^{u}$ with endpoints in $V_{u,i}$ . Since $a_{i}$ s are pairwise differently and are contained in $R(u)$ , and $H^{u}$ has at most $\lfloor|W^{u}|/2\rfloor$ Hamiltonian paths to afford by Corollary 1.4, we only need to ensure that $|R(u)|+\lfloor|W^{u}|/2\rfloor≥ n_{1}(u)+n_{3}(u)$ . Thus, we only need to prove the following result.
**Lemma 2.1**
*For each labelled vertex $u∈ V(G_{s})$ with $|W^{u}|≥ 2$ , Ineq.
$$
\displaystyle n_{1}(u)+n_{3}(u)-|R(u)|\leq\lfloor|W^{u}|/2\rfloor \tag{4}
$$
holds.*
* Proof*
By Eqs. (2) and (3),
| | $\displaystyle n_{1}(u)+n_{3}(u)-|R(u)|$ | $\displaystyle=n_{1}(u)+n_{3}(u)-[\ell-|W^{u}|-n_{2}(u)-2n_{4}(u)-n_{5}(u)]$ | |
| --- | --- | --- | --- |
Since $c=\ell-\lceil k/2\rceil$ , it follows that
$$
\displaystyle n_{1}(u)+n_{3}(u)-|R(u)|=n_{4}(u)+|W^{u}|-\lceil k/2\rceil. \tag{5}
$$ Before the proof of Lemma 2.1, we give a series of claims as preliminaries.
**Claim 2**
*Suppose that $a∈ V(v_{root}\overrightarrow{T}u)$ is a labelled vertex with $a∈ U^{\iota}$ , where $1≤\iota≤ n$ . If $|W^{a}|=1$ (say $W^{a}=\{x\}$ ), then $n_{5}(a)≤ 1$ . Moreover,
1. if $n_{5}(a)=1$ , then $n_{4}(x)≤ 1$ ;
1. if $n_{5}(a)=0$ , then $n_{3}(x)=n_{4}(x)=n_{5}(x)=0$ .*
* Proof*
Note that $x∈ U^{\iota-1}$ . in Algorithm 1 (Lines 18–39), if $T_{i}^{\iota}$ is not a tree of Type 5 on $a$ , then $F_{i}^{a}$ is chosen such that $d_{T_{i}^{\iota-1}}(x)=1$ (here $W^{a}=\{x\}$ ); if $T^{\iota}_{i}$ is a tree of Type 5 on $a$ , then $F_{i}^{a}$ is chosen such that $d_{T_{i}^{\iota-1}}(x)=2$ . Since $|W^{a}|=1$ , there is at most one $T^{\iota}_{i}$ of Type 5 on $a$ , and hence $n_{5}(a)≤ 1$ . Furthermore, if $n_{5}(a)=1$ , then there is exact one $T_{i}^{\iota-1}$ in $G^{\ell-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ . Hence, $n_{4}(x)≤ 1$ . If $n_{5}(a)=0$ , then $d_{T_{i}^{\iota-1}}(x)=1$ for each $T_{i}^{\iota-1}$ . Therefore, $n_{3}(x)=n_{4}(x)=n_{5}(x)=0$ . ∎
**Claim 3**
*Suppose that $a,b$ are two labelled vertices of $v_{root}\overrightarrow{T}u$ and $a∈ W^{b}$ , where $a∈ U^{\iota}$ for some $\iota∈[n]$ . Then there are at most $\max\{1,n_{1}(b)+n_{3}(b)-|R(b)|+1\}$ $T_{i}^{\iota}$ s such that $d_{T_{i}^{\iota}}(a)=2$ . Furthermore, $n_{4}(a)≤\max\{1,n_{1}(b)+n_{3}(b)-|R(b)|+1\}$ .*
* Proof*
Since $a∈ W^{b}$ and $a∈ U^{\iota}$ , it follows that $b∈ U^{\iota+1}$ . For an $i∈[c]$ with $d_{T_{i}^{\iota}}(a)=2$ , we have that either $F_{i}^{b}$ is a Hamiltonian path of the clique $H^{b}[W^{b}]$ , or $a∈ V_{b,i}\cap W^{b}$ for some $i∈[c]$ (recall that $V_{b,i}=\{z∈ V(H^{b}):z\mbox{ is an end vertex of some edge in }E_{b,i}\}$ , where $E_{b,i}$ is the set of edges in $T_{i}^{\iota+1}$ incident with $b$ ). Hence, if there are $h$ edge-disjoint $F_{i}^{b}$ s that are constructed as Hamiltonian paths of $H^{b}[W^{b}]$ , then there are at most $h+1$ internally disjoint $U^{\iota}$ -Steiner trees $T_{i}^{\iota}$ such that $d_{T_{i}^{\iota}}(a)=2$ . By the definition of $n_{4}(a)$ , we have that $n_{4}(a)≤ h+1$ . In Algorithm 1 (Lines 28–36), if $n_{1}(b)+n_{3}(b)-|R(b)|>0$ , then there are at most $n_{1}(b)+n_{3}(b)-|R(b)|$ $F_{i}^{b}$ s that are constructed as Hamiltonian paths in $H^{b}[W^{b}]$ ; if $n_{1}(b)+n_{3}(b)-|R(b)|≤ 0$ , there is no $F_{i}^{b}$ that is constructed as a Hamiltonian path in $H^{b}[W^{b}]$ . Hence, $h≤\max\{0,n_{1}(b)+n_{3}(b)-|R(b)|\}$ . Thus, there are at most $\max\{1,n_{1}(b)+n_{3}(b)-|R(b)|+1\}$ internally disjoint $U^{\iota}$ -Steiner trees $T_{i}^{\iota}$ such that $d_{T_{i}^{\iota}}(a)=2$ , and $n_{4}(a)≤\max\{1,n_{1}(b)+n_{3}(b)-|R(b)|+1\}$ . ∎
**Claim 4**
*Let $a∈ V(v_{root}\overrightarrow{T}u)$ be a labelled vertex, where $a∈ U^{\iota}$ for some $\iota∈[n]$ . If $n_{4}(a)≤ 1$ and $2≤|W^{a}|≤\left\lceil k/2\right\rceil-1$ , then $n_{1}(a)+n_{3}(a)-|R(a)|≤ 0$ . Moreover, for each vertex $x∈ W^{a}$ , there is at most one $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ .*
* Proof*
Since $n_{4}(a)≤ 1$ and $|W^{a}|≤\left\lceil k/2\right\rceil-1$ , it follows from Eq. (5) that $n_{1}(a)+n_{3}(a)-|R(a)|=|W^{a}|-\left\lceil k/2\right\rceil+1≤ 0.$ By Claim 3, for each vertex $x∈ W^{a}$ , there is at most one $T_{i}^{a}$ such that $d_{T_{i}^{\iota-1}}(x)≤ 2$ . ∎
**Claim 5**
*Let $a∈ V(v_{root}\overrightarrow{T}u)$ be a labelled vertex, where $a∈ U^{\iota}$ for some $\iota∈[n]$ . If $n_{4}(a)≤ 1$ and $\left\lceil k/2\right\rceil≤|W^{a}|≤ k-1$ , then $n_{1}(a)+n_{3}(a)-|R(a)|≤\lfloor|W^{a}|/2\rfloor$ . Moreover, for each $x∈ W^{a}$ ,
1. if $n_{4}(a)=1$ , then there are at most $|W^{a}|-\left\lceil k/2\right\rceil+2$ internally disjoint $U^{\iota-1}$ -Steiner trees $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ ;
1. if $n_{4}(a)=0$ , then there are at most $|W^{a}|-\left\lceil k/2\right\rceil+1$ internally disjoint $U^{\iota-1}$ -Steiner trees $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ .*
* Proof*
Since $n_{4}(a)≤ 1$ , it follows from Eq. (5) that
| | $\displaystyle n_{1}(a)+n_{3}(a)-|R(a)|$ | $\displaystyle≤|W^{a}|-\left\lceil k/2\right\rceil+n_{4}(a)$ | |
| --- | --- | --- | --- |
and the equality indicates $n_{4}(a)=1$ . If $|W^{a}|≤ k-2$ , then
$$
n_{1}(a)+n_{3}(a)-|R(a)|\leq|W^{a}|-\left\lceil(|W^{a}|+2)/2\right\rceil+1\leq%
\left\lfloor|W^{a}|/2\right\rfloor;
$$
if $|W^{a}|=k-1$ and $k$ is even, then
$$
n_{1}(a)+n_{3}(a)-|R(a)|=|W^{a}|-\left\lceil(|W^{a}|+1)/2\right\rceil+1=\left%
\lfloor(|W^{a}|+1)/2\right\rfloor=\left\lfloor|W^{a}|/2\right\rfloor;
$$
if $|W^{a}|=k-1$ and $k$ is odd, then
$$
n_{1}(a)+n_{3}(a)-|R(a)|=(k-1)-\left\lceil k/2\right\rceil+1\leq\left\lfloor k%
/2\right\rfloor=\left\lfloor(k-1)/2\right\rfloor=\left\lfloor|W^{a}|/2\right\rfloor.
$$
Therefore, $n_{1}(a)+n_{3}(a)-|R(a)|≤\left\lfloor|W^{a}|/2\right\rfloor$ . By Eq. (5) and Claim 3, if $n_{4}(a)=1$ , then for each vertex $x∈ W^{a}$ , there are at most $n_{1}(a)+n_{3}(a)-|R(a)|+1=|W^{a}|-\left\lceil k/2\right\rceil+2$ internally disjoint $U^{\iota-1}$ -Steiner trees $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ ; if $n_{4}(a)=0$ , then for each vertex $x∈ W^{a}$ , there are at most $n_{1}(a)+n_{3}(a)-|R(a)|+1=|W^{u}|-\left\lceil k/2\right\rceil+1$ internally disjoint $U^{\iota-1}$ -Steiner trees $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ . ∎
**Claim 6**
*Suppose $a,b∈ V(\overrightarrow{T})$ , $a\prec b$ and $a\overrightarrow{T}b=az_{1}z_{2}... z_{p}b$ , where $a∈ U^{\iota}$ for some $\iota∈[n]$ . If $|W^{z_{i}}|≤\lceil k/2\rceil-1$ for each $i∈[p]$ , and either $|W^{a}|=1$ or $2≤|W^{a}|≤\lceil k/2\rceil-1$ and $n_{4}(a)≤ 1$ , then $n_{4}(b)≤ 1$ .*
* Proof*
Since $a∈ U^{\iota}$ , it follows that $z_{q}∈ U^{\iota-q}$ for each $q∈[p]$ and $b∈ U^{\iota-p-1}$ . Since $|W^{a}|=1$ or $2≤|W^{a}|≤\lceil k/2\rceil-1$ and $n_{4}(a)≤ 1$ , by Claims 2 and 4, there are at most one $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(z_{1})=2$ . Hence, $n_{4}(z_{1})≤ 1$ . Since $|W^{z_{1}}|≤\lceil k/2\rceil-1$ and $n_{4}(z_{1})≤ 1$ , by Claims 2 and 4, there are at most one $T_{i}^{\iota-2}$ such that $d_{T_{i}^{\iota-2}}(z_{2})=2$ . Hence, $n_{4}(z_{2})≤ 1$ . Repeat this progress, we can get that $n_{4}(z_{p})≤ 1$ . Since $|W^{z_{p}}|≤\lceil k/2\rceil-1$ and $n_{4}(z_{p})≤ 1$ , by Claims 2 and 4, there are at most one $T_{i}^{\iota-p-1}$ such that $d_{T_{i}^{\iota-p-1}}(b)=2$ . Hence, $n_{4}(b)≤ 1$ . ∎
**Claim 7**
*Suppose $|W^{v_{root}}|≤\lceil k/2\rceil$ , $b∈ V(\overrightarrow{T})$ and $v_{root}\overrightarrow{T}b=v_{root}z_{1}z_{2}... z_{p}b$ . If $|W^{z_{i}}|=1$ for each $i∈[p]$ , then $n_{4}(b)=0$ .*
* Proof*
Since $|W^{v_{root}}|≤\lceil k/2\rceil$ , by Claim 1, $d_{T_{i}^{n-1}}(z_{1})=1$ for each $U^{n-1}$ -Steiner tree $T_{i}^{n-1}$ . Hence, $n_{3}(z_{1})=n_{4}(z_{1})=n_{5}(z_{1})=0$ . Since $|W^{z_{1}}|=1$ and $n_{5}(z_{1})=0$ , by the second statement of Claim 2, $n_{3}(z_{2})=n_{4}(z_{2})=n_{5}(z_{2})=0$ . Since $|W^{z_{2}}|=1$ and $n_{5}(z_{2})=0$ , by the second statement of Claim 2, $n_{3}(z_{3})=n_{4}(z_{3})=n_{5}(z_{3})=0$ . Repeat this process, we get that $n_{4}(b)=0$ . ∎
**Claim 8**
*Suppose that $a,b$ are two labelled vertices and $a∈ W^{b}$ , where $a∈ U^{\iota}$ for some $\iota∈[n]$ . If $k$ is even and $|W^{b}|=|W^{a}|=k/2$ , then $n_{1}(a)+n_{3}(a)-|R(a)|≤\lfloor|W^{a}|/2\rfloor$ and the following hold.
1. If $b=v_{root}$ , then for each vertex $x∈ W^{a}$ , there is at most one $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ .
1. If $b≠ v_{root}$ , then for each vertex $x∈ W^{a}$ , there are at most two $T_{i}^{\iota-1}$ s such that $d_{T_{i}^{\iota-1}}(x)=2$ .*
* Proof*
Suppose $b=v_{root}$ . Then by Claim 1, $n_{4}(a)=0$ . Thus,
$$
n_{1}(a)+n_{3}(a)-|R(a)|=n_{4}(a)+|W^{a}|-k/2=|W^{a}|-k/2=0.
$$
By Claim 3, for each vertex $x∈ W^{a}$ , there is at most one $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ . Now assume that $b≠ v_{root}$ . Without loss of generality, suppose $v_{root}\overrightarrow{T}b=v_{root}v_{1}v_{2}... v_{p}b$ . By Fact 1, we have that $|W^{root}|=2≤ k/2$ and $|W^{v_{i}}|=1$ for each $i∈[p]$ . By Claim 7, $n_{4}(b)=0$ . By Claim 3,
| | $\displaystyle n_{4}(a)$ | $\displaystyle≤\max\{1,n_{1}(b)+n_{3}(b)-|R(b)|+1\}$ | |
| --- | --- | --- | --- |
By Claim 3 again, for each vertex $x∈ W^{a}$ , there are at most two $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ . ∎
With the above preparations, we now prove Lemma 2.1. Recall that $u∈ V(G_{s})$ is a labelled vertex with $|W^{u}|≥ 2$ . Then each vertex of $V(v_{root}\overrightarrow{T}u)$ is also a labelled vertex. Suppose that $v^{*}$ is the maximum vertex of $v_{root}\overrightarrow{T}u$ such that one of the following holds (if such vertex $v^{*}$ exists).
1. $|W^{v^{*}}|=1$ ,
1. $2≤|W^{v^{*}}|≤\lceil k/2\rceil-1$ and $n_{4}(v^{*})≤ 1$ .
We distinguish the following two cases to show this lemma, that is, to prove Ineq. (4) holds (recall that the Ineq. (4) is $n_{1}(u)+n_{3}(u)-|R(u)|≤\lfloor|W^{u}|/2\rfloor$ ).
**Case 1**
*$v^{*}$ exists.*
Let $\overrightarrow{P}=v^{*}\overrightarrow{T}u=v^{*}z_{1}z_{2}... z_{p}u$ . By the maximality of $v^{*}$ , we have that $|W^{z_{i}}|≥ 2$ for each $i∈[p]$ . If $|\overrightarrow{P}|=1$ , then $v^{*}=u$ . Since $|W^{u}|≥ 2$ , it follows from ( $ii$ ) that $2≤|W^{u}|≤\lceil k/2\rceil-1$ and $n_{4}(u)≤ 1$ . By Claim 4, we have that $n_{1}(u)+n_{3}(u)-|R(u)|≤ 0$ , and Ineq. (4) holds. Thus, we assume that $|\overrightarrow{P}|≥ 2$ below. By Claim 6, we have that
$$
\displaystyle n_{4}(z_{1})\leq 1. \tag{6}
$$
Thus, by the maximality of $v^{*}$ , we have that $|W^{z_{1}}|≥\lceil k/2\rceil$ . Since $|W^{v_{root}}|+|W^{z_{1}}|≤ k+1$ (by Fact 1) and $|W^{v_{root}}|≥ 2$ , it follows that $|W^{z_{1}}|≤ k-1$ . Hence,
$$
\displaystyle\lceil k/2\rceil\leq|W^{z_{1}}|\leq k-1. \tag{7}
$$
**Subcase 1.1**
*$p=0$ .*
Then $\overrightarrow{P}=v^{*}\overrightarrow{T}u=v^{*}u$ and $z_{1}=u$ . Since $n_{4}(u)≤ 1$ and $\lceil k/2\rceil≤|W^{u}|≤ k-1$ by Ineqs. (6) and (7), it follows from Claim 5 that $n_{1}(u)+n_{3}(u)-|R(u)|≤\lfloor|W^{u}|/2\rfloor$ , Ineq. (4) holds.
**Subcase 1.2**
*$p=1$ .*
Then $\overrightarrow{P}=v^{*}\overrightarrow{T}u=v^{*}z_{1}u$ and $u=z_{2}$ . By Fact 1, we have that
$$
\displaystyle|W^{z_{1}}|+|W^{u}|+|W^{v_{root}}|-2\leq k. \tag{8}
$$
Hence $|W^{z_{1}}|+|W^{u}|≤ k$ . Recall that $|W^{z_{1}}|≥\lceil k/2\rceil$ . If $|W^{u}|≥\lceil k/2\rceil$ , then $k$ is even and $|W^{z_{1}}|=|W^{u}|=k/2$ . By Claim 8, we have that $n_{1}(u)+n_{3}(u)-|R(u)|≤\lfloor|W^{u}|/2\rfloor$ . Therefore, Ineq. (4) holds. Now, we assume that $2≤|W^{u}|≤\lceil k/2\rceil-1$ . Since $n_{4}(z_{1})≤ 1$ and $k-1≥|W^{z_{1}}|≥\lceil k/2\rceil$ , by Claim 5, there are at most $|W^{z_{1}}|-\left\lceil k/2\right\rceil+2$ internally disjoint $U^{s}$ -Steiner trees $T_{i}^{s}$ such that $d_{T_{i}^{s}}(u)=2$ (note that $u∈ G_{s}$ ). Hence, $n_{4}(u)≤|W^{z_{1}}|-\left\lceil k/2\right\rceil+2$ . Thus, by Eq. (5),
$$
n_{1}(u)+n_{3}(u)-|R(u)|=n_{4}(u)+|W^{u}|-\left\lceil k/2\right\rceil\leq|W^{u%
}|+|W^{z_{1}}|-2\left\lceil k/2\right\rceil+2.
$$
If $|W^{u}|+|W^{z_{1}}|≤ k-1$ or $k$ is odd, then $n_{1}(u)+n_{3}(u)-|R(u)|≤ 1≤\lfloor|W^{u}|/2\rfloor$ , Ineq. (4) holds. Thus, assume that $|W^{u}|+|W^{z_{1}}|=k$ (recall that $|W^{u}|+|W^{z_{1}}|≤ k$ ) and $k$ is even below ( $k≥ 4$ ). Since $|W^{z_{1}}|≥\lceil k/2\rceil$ , it follows that $|W^{u}|≤ k/2$ . Suppose that $v_{root}\overrightarrow{T}z_{p}=v_{root}w_{1}w_{2}... w_{q}v^{*}z_{p}$ . Since $|W^{v_{root}}|≥ 2$ , it follows from Ineq. (8) and Fact 1 that $|W^{v_{root}}|=2≤ k/2$ and $|W^{w_{1}}|=...=|W^{w_{q}}|=|W^{v^{*}}|=1$ . According to Claim 7, we have that $n_{4}(u)=0$ . Hence,
| | $\displaystyle n_{1}(u)+n_{3}(u)-|R(u)|$ | $\displaystyle=n_{4}(u)+|W^{u}|-\left\lceil k/2\right\rceil≤ 0≤\lfloor|W^%
{u}|/2\rfloor,$ | |
| --- | --- | --- | --- |
the Ineq. (4) holds.
**Subcase 1.3**
*$p≥ 2$ .*
Then $u\succeq z_{3}$ . Recall that $|W^{z_{i}}|≥ 2$ for each $i∈[p]$ . Since
$$
\displaystyle|W^{v_{root}}|+|W^{z_{1}}|+|W^{z_{2}}|+|W^{z_{3}}|-3\leq k \tag{9}
$$
and $|W^{z_{1}}|≥\lceil k/2\rceil$ , it follows that $|W^{z_{2}}|,|W^{z_{3}}|≤\lfloor k/2\rfloor-1$ and $|W^{z_{1}}|+|W^{z_{2}}|≤ k-1$ . On the other hand, since $|W^{z_{3}}|≥ 2$ , it follows that $k≥ 6$ . Without loss of generality, suppose $z_{2}∈ U^{\iota}$ . Recall Ineqs. (6) and (7), and combine with Claim 5, there are at most $|W^{z_{1}}|-\left\lceil\frac{k}{2}\right\rceil+2$ internally disjoint $U^{\iota}$ -Steiner trees $T_{i}^{\iota}$ such that $d_{T_{i}^{\iota}}(z_{2})=2$ . Hence, $n_{4}(z_{2})≤|W^{z_{1}}|-\left\lceil\frac{k}{2}\right\rceil+2$ . By Claim 3, $n_{4}(z_{3})≤\max\{1,n_{4}(z_{2})+|W^{z_{2}}|-\lceil k/2\rceil+1\}$ . However, by the maximality of $v^{*}$ , we have that $n_{4}(z_{3})≥ 2$ . Hence,
$$
\displaystyle n_{4}(z_{3})\leq n_{4}(z_{2})+|W^{z_{2}}|-\lceil k/2\rceil+1\leq%
|W^{z_{1}}|+|W^{z_{2}}|-2\lceil k/2\rceil+3. \tag{10}
$$
Since $|W^{z_{1}}|+|W^{z_{2}}|≤ k-1$ , it follows that if $k$ is odd or $|W^{z_{1}}|+|W^{z_{2}}|≤ k-2$ , then $n_{4}(z_{3})≤ 1$ , a contradiction. Hence, $k$ is even and $|W^{z_{1}}|+|W^{z_{2}}|=k-1$ . This implies that $|W^{z_{3}}|=n_{4}(z_{3})=2$ by Ineqs. (9) and (10). Since $k≥ 6$ , it follows that $n_{4}(z_{3})+|W^{z_{3}}|-\lceil k/2\rceil≤ 1≤\lfloor|W^{z_{3}}|/2\rfloor$ . Recall that $u\succeq z_{3}$ . If $v_{3}=u$ , then
$$
n_{1}(u)+n_{3}(u)-|R(u)|=n_{4}(u)+|W^{u}|-\lceil k/2\rceil\leq\lfloor|W^{u}|/2\rfloor,
$$
Ineq. (4) holds. If $u\succ z_{3}$ , then since
| | $\displaystyle|W^{v_{root}}|+|W^{z_{1}}|+|W^{z_{2}}|+|W^{z_{3}}|+|W^{u}|-4≤ k,$ | |
| --- | --- | --- |
we have that
| | $\displaystyle|W^{u}|≤ k+4-(|W^{v_{root}}|+|W^{z_{1}}|+|W^{z_{2}}|+|W^{z_{3}%
}|)=1,$ | |
| --- | --- | --- |
a contradiction.
**Case 2**
*$v^{*}$ does not exist.*
Let $\overrightarrow{P}=v_{root}\overrightarrow{T}u=v_{root}z_{1}...,z_{t}u$ . Since $v^{*}$ does not exist, for each vertex $z∈ V(\overrightarrow{P})$ , either $2≤|W^{z}|≤\lceil k/2\rceil-1$ and $n_{4}(z)≥ 2$ , or $|W^{z}|≥\lceil k/2\rceil$ . Since $n_{4}(v_{root})=0$ , it follows that $|W^{v_{root}}|≥\lceil k/2\rceil$ . By Claim 1, $n_{4}(z_{1})≤|W^{v_{root}}|-\lceil k/2\rceil$ .
**Subcase 2.1**
*$|W^{z_{1}}|≥\lceil k/2\rceil$ .*
Since $|W^{v_{root}}|,|W^{z_{1}}|≥\lceil k/2\rceil$ , we have that $t≤ 1$ . Otherwise,
$$
\sum_{z\in V(\overrightarrow{P})}|W^{z}|-(|\overrightarrow{P}|-1)>k,
$$
which contradicts Fact 1. Moreover, if $t=1$ , then $k$ is even, $|W^{v_{root}}|=|W^{z_{1}}|=k/2$ and $|W^{u}|=2$ . Suppose that $t=0$ . Then $u=z_{1}$ , and hence $n_{4}(u)≤|W^{v_{root}}|-\lceil k/2\rceil$ . Thus
| | $\displaystyle n_{1}(u)+n_{3}(u)-|R(u)|$ | $\displaystyle=n_{4}(u)+|W^{u}|-\lceil k/2\rceil$ | |
| --- | --- | --- | --- |
Ineq. (4) holds. Suppose $t=1$ . Then $\overrightarrow{P}=v_{root}z_{1}u$ . Hence, $k$ is even ( $k≥ 4$ ), $|W^{v_{root}}|=|W^{z_{1}}|=k/2$ and $|W^{u}|=2$ . By the first statement of Claim 8 (here, we regard $v_{root}$ and $z_{1}$ as the vertices $b$ and $a$ in Claim 8, respectively, and then $u$ can be regarded as the vertex $x$ in Claim 8), we have that $n_{4}(u)≤ 1$ . Hence,
$$
n_{1}(u)+n_{3}(u)-|R(u)|=n_{4}(u)+|W^{u}|-\lceil k/2\rceil\leq 1\leq\lfloor|W^%
{u}|/2\rfloor,
$$
Ineq. (4) holds.
**Subcase 2.2**
*$2≤|W^{z_{1}}|≤\lceil k/2\rceil-1$ .*
Recall that $n_{4}(z_{1})=|W^{v_{root}}|-\lceil k/2\rceil$ . Since $n_{4}(z_{1})≥ 2$ , $|W^{v_{root}}|≥\lceil k/2\rceil+2$ . If $|W^{v_{root}}|+|W^{z_{1}}|=k+1$ , then $v=v_{root}$ , $u=z_{1}$ and
$$
n_{1}(u)+n_{3}(u)-|R(u)|=n_{4}(u)+|W^{u}|-\lceil k/2\rceil\leq 1\leq\lfloor|W^%
{u}|/2\rfloor.
$$
Thus, suppose $|W^{v_{root}}|+|W^{z_{1}}|≤ k$ . Then
$$
n_{1}(z_{1})+n_{3}(z_{1})-|R(z_{1})|=n_{4}(z_{1})+|W^{z_{1}}|-\lceil k/2\rceil%
\leq 0.
$$
If $z_{1}=u$ , then $n_{1}(u)+n_{3}(u)-|R(u)|≤ 0,$ Ineq. (4) holds. Now, assume that $z_{1}≠ u$ . Then $z_{2}$ exists. By Claim 3, $n_{4}(z_{2})≤ 1$ . Since $z_{2}$ is not a candidate of $v^{*}$ , it follows that $|W^{z_{2}}|≥\lceil k/2\rceil$ . Since $|W^{v_{root}}|≥\lceil k/2\rceil+2,|W^{z_{2}}|≥\lceil k/2\rceil$ and $|W^{z_{1}}|≥ 2$ , it follows that $|W^{v_{root}}|+|W^{z_{1}}|+|W^{z_{2}}|-2≥ k+2$ , which contradicts Fact 1. ∎
With the conclusion of Lemma 2.1, the proof of Theorem 2.1 is completed.
2.3 Proof of Theorem 1.5
We first consider $3≤ k≤\ell$ . The lower bound $\ell-\lceil k/2\rceil≤\kappa_{k}(S(n,\ell))≤\lambda_{k}(S(n,\ell))$ can be obtained from Theorem 2.1 directly. For the upper bounds of $\kappa_{k}(S(n,\ell))$ and $\lambda_{k}(S(n,\ell))$ , consider the graph $G_{n-1}$ . Let $V(G_{n-1})=\{u_{1},...,u_{\ell}\}$ , $U=\{x_{1},...,x_{k}\}$ and $x_{i}∈ A_{n-1,u_{i}}$ , where $k≤\ell$ . Suppose there are $p$ edge-disjoint $U$ -Steiner trees $T^{\prime}_{1},...,T^{\prime}_{p}$ of $G=S(n,\ell)$ and $\mathcal{P}=\{A_{n-1,u}:u∈ V(G_{n-1})\}$ . Let $T^{*}_{i}=T^{\prime}_{i}/\mathcal{P}$ for $i∈[p]$ . Then $T^{*}_{1},...,T^{*}_{p}$ are edge-disjoint connected graphs of $G_{n-1}$ containing $\{u_{1},...,u_{k}\}$ . Thus, $p≤\lambda_{k}(G_{n-1})=\lambda_{k}(K_{\ell})=\ell-\lceil k/2\rceil$ . Therefore, $\kappa_{k}(S(n,\ell))≤\lambda_{k}(S(n,\ell))≤\ell-\lceil k/2\rceil$ , the upper bound follows.
Now consider the case $\ell+1≤ k≤{\ell}^{n}$ . By Theorem 1.2, $\lambda_{k}(S(n,\ell))≥\lfloor\ell/2\rfloor$ . Since $\lambda_{k}(S(n,\ell))≤\lambda_{\ell}(S(n,\ell))=\lfloor\ell/2\rfloor$ , it follows that $\kappa_{k}(S(n,\ell))≤\lambda_{k}(S(n,\ell))≤\lfloor\ell/2\rfloor$ . Therefore, $\lambda_{k}(S(n,\ell))=\lfloor\ell/2\rfloor$ and $\kappa_{k}(S(n,\ell))≤\lfloor\ell/2\rfloor$ . The proof is completed.
3 Some network properties
Generalized connectivity is a graph parameter to measure the stability of networks. In the following part, we will give the following other properties of Sierpiński graphs.
The Sierpiński graph(networks) is obtained after $t$ iteration as $S(t,\ell)$ that has $N_{t}$ nodes and $E_{t}$ edges, where $t=0,1,2,·s,T-1$ , and $T$ is the total number of iterations, and our generation process can be illustrated as follows.
<details>
<summary>x3.png Details</summary>
### Visual Description
## Line Chart: Exponential Growth Comparison
### Overview
The image is a line chart comparing the exponential growth of two functions: 3^t and (1/2)*(3^(t+1) - 3). The x-axis represents the variable 't', and the y-axis represents the function's value. Both functions exhibit exponential growth, but the second function, (1/2)*(3^(t+1) - 3), grows slightly faster.
### Components/Axes
* **X-axis:**
* Label: No explicit label, but the axis represents the variable 't'.
* Scale: 1 to 14, with tick marks at each integer value.
* **Y-axis:**
* Label: No explicit label, but the axis represents the function's value.
* Scale: 0 to 2.5 x 10^6 (2,500,000), with tick marks at 500,000 intervals.
* **Legend:** Located in the top-right corner.
* Blue line: Represents the function 3^t.
* Gold line: Represents the function (1/2)*(3^(t+1) - 3).
### Detailed Analysis
* **Blue Line (3^t):**
* Trend: Exponential growth. Initially, the value is close to zero, but it increases rapidly after t=10.
* Data Points:
* t=1: Approximately 3
* t=5: Approximately 243
* t=10: Approximately 59,049
* t=12: Approximately 531,441
* t=14: Approximately 4,782,969 (extrapolated)
* **Gold Line ((1/2)*(3^(t+1) - 3)):**
* Trend: Exponential growth, slightly faster than the blue line.
* Data Points:
* t=1: Approximately 3
* t=5: Approximately 360
* t=10: Approximately 88,572
* t=12: Approximately 797,160
* t=14: Approximately 7,174,452 (extrapolated)
### Key Observations
* Both functions exhibit exponential growth, but the gold line function grows faster than the blue line function.
* The difference between the two functions becomes more pronounced as 't' increases.
* For t < 8, both functions are close to zero.
### Interpretation
The chart visually demonstrates the power of exponential growth. Even with a relatively small difference in the base or exponent, the resulting values diverge significantly as the variable 't' increases. The function (1/2)*(3^(t+1) - 3) grows faster than 3^t, indicating that even slight modifications to an exponential function can lead to substantial changes in its behavior over time. This has implications in various fields, such as finance (compound interest), biology (population growth), and computer science (algorithm complexity).
</details>
Figure 7: The Function of $E_{t}=3^{t}$ and $N_{t}=\frac{3^{t+1}-3}{2}$
Step 1: Initialization. Set $t=0$ , $G_{1}$ is a complete graph of order $\ell$ , and thus $N_{1}=\ell$ and $E_{1}=\binom{\ell}{2}$ . Set $G_{1}=S(1,\ell)$ .
Step 2: Generation of $G_{t+1}$ from $G_{t}$ . Let $S^{1}(t,\ell),S^{2}(t,\ell),...,S^{\ell}(t,\ell)$ be all Sierpiński graphs added at Step $t$ , where $S^{i}(t,\ell)\cong S(t,\ell)$ ( $1≤ i≤\ell$ ). At Step $t+1$ , we add one edge (bridge edge) between $S^{i}(t,\ell)$ and $S^{j}(t,\ell)$ , $i≠ j$ , namely the edge between vertices $\langle ij·s j\rangle$ and $\langle ji·s i\rangle$ .
For Sierpiński graphs $G_{t}=S(t,\ell)$ , its order and size are $N_{t}=\ell^{t}$ and $E_{t}=\frac{\ell^{t+1}-\ell}{2}$ , respectively; see Table 2 and Figure 7 (for $\ell=3$ ).
Table 2: The size and order of $G_{t}$ for $\ell=3$
| $t$ | 1 | 2 | 3 | 4 | 5 | 6 | t |
| --- | --- | --- | --- | --- | --- | --- | --- |
| $N_{t}$ | 3 | 9 | 27 | 81 | 242 | 729 | $\ell^{t}$ |
| $E_{t}$ | 3 | 12 | 39 | 120 | 363 | 1092 | $\frac{\ell^{t+1}-\ell}{2}$ |
The degree distribution for $t$ times are
$$
\left(\underbrace{\ell-1,\ell-1,\cdots,\ell-1}_{\ell},\underbrace{\ell,\ell,%
\ell,\cdots,\ell}_{\ell^{t}-\ell}\right). \tag{11}
$$
From Equation 11, the instantaneous degree distribution is $P(\ell-1,t)=1/\ell^{t-1}$ for $t=2,·s,T$ and $P(\ell,t)=(\ell^{t}-\ell)/\ell^{t}$ for $t=2,·s,T$ . Note that the density of Sierpiński graphs is $\rho=E_{t}/\binom{N_{t}}{2}→ 0$ for $t→+∞$ . For large enough $\ell$ and any $1≤ k≤\ell$ , we have $|\{v∈ V(G)|d_{S(t,\ell)}(v)|≥ k\}≈|V(S(t,\ell))|$ .
**Theorem 3.1**
*[11] If $n∈\mathbb{N}$ and $G$ is a graph, then $\kappa\left(S(n,G)\right)=\kappa(G)$ and $\lambda\left(S(n,G)\right)=\lambda(G)$ .*
From Theorem 3.1, we have $\kappa\left(S(n,\ell)\right)=\kappa(K_{\ell})=\ell-1$ and $\lambda\left(S(n,\ell)\right)=\lambda(K_{\ell})=\ell-1$ . Note that $\lambda_{k}(S(n,\ell))=\ell-\left\lceil k/2\right\rceil$ and $\kappa_{k}(S(n,\ell))=\ell-\left\lceil k/2\right\rceil$ ; see Figure 8.
<details>
<summary>x4.png Details</summary>
### Visual Description
## 3D Scatter Plot: l, k, and l - floor(k/2)
### Overview
The image is a 3D scatter plot visualizing the relationship between three variables: `l`, `k`, and `l - floor(k/2)`. The plot displays data points for each variable combination, with `l` and `k` ranging from approximately 0 to 30. The plot uses color-coding to distinguish the three variables: blue for `l - floor(k/2)`, orange for `l`, and green for `k`.
### Components/Axes
* **X-axis:** Ranges from 0 to 30, unlabeled.
* **Y-axis:** Ranges from 0 to 30, unlabeled.
* **Z-axis:** Ranges from -10 to 30, unlabeled.
* **Legend (Top-Right):**
* Blue square: `l - floor(k/2)`
* Orange square: `l`
* Green square: `k`
### Detailed Analysis
The data points are arranged in a grid-like structure, suggesting a systematic variation of `l` and `k`.
* **Green (`k`):** The green data points form a flat plane at the top of the plot, indicating that the z-value of these points corresponds directly to the value of `k`. The z-value of the green points is equal to the y-axis value.
* **Orange (`l`):** The orange data points form a plane that slopes upward as both x and y increase. The z-value of the orange points is equal to the x-axis value.
* **Blue (`l - floor(k/2)`):** The blue data points form a more complex surface. For a fixed value of `l`, the z-value decreases as `k` increases. The blue points are always below the orange points.
### Key Observations
* The `k` values (green) are consistently the highest, forming the upper bound of the data.
* The `l` values (orange) are intermediate, forming a plane below the `k` values.
* The `l - floor(k/2)` values (blue) are the lowest, and their distribution is influenced by the floor function, creating a stepped pattern.
### Interpretation
The plot visualizes the relationship between `l`, `k`, and the expression `l - floor(k/2)`. The plot demonstrates that `k` is independent of `l`, and `l` is independent of `k`. The expression `l - floor(k/2)` shows how the value of `l` is reduced based on the value of `k`. The floor function introduces a discrete, stepped behavior to the `l - floor(k/2)` values. The plot is useful for understanding how the value of `l - floor(k/2)` changes as `l` and `k` vary.
</details>
Figure 8: The generalized (edge-)connectivity of $S(n,\ell)$
The number of spanning tree of $G$ denoted by $\tau(G)$ . Let
$$
\rho(G)=\lim_{V(G)\longrightarrow\infty}\frac{\ln|\tau(G)|}{\left|V(G)\right|}, \tag{12}
$$
where $\rho(G)$ is called the entropy of spanning trees or the asymptotic complexity [2, 7].
As an application of generalized (edge-)connectivity, similarly to the Equation 12, it can describe the fault tolerance of a graph or network, a common metric is called the entropy of spanning trees.
we give the entropy of the $k$ -Steiner tree of a graph $G$ can be defined as
$$
\rho_{k}(G)=\lim_{|V(G)|\longrightarrow\infty}\frac{\ln|\kappa_{k}(G)|}{\left|%
V(G)\right|}
$$
The entropy of the $3,6,9$ -Steiner tree of Sierpiński graph $S(8,\ell)$ can be seen in Figure 9.
<details>
<summary>x5.png Details</summary>
### Visual Description
## Line Chart: Logarithmic Plots
### Overview
The image is a line chart displaying three logarithmic functions. The x-axis ranges from approximately 3 to 9, and the y-axis ranges from -2e-6 to 4e-6. The chart plots three different logarithmic functions, each represented by a different color: blue, orange, and green.
### Components/Axes
* **X-axis:** Ranges from 3 to 9, with integer markers.
* **Y-axis:** Ranges from -2 x 10^-6 to 4 x 10^-6, with markers at -2 x 10^-6, 0, 2 x 10^-6, and 4 x 10^-6.
* **Legend (Top-Right):**
* Blue line: log(|-3/2|)/l^8
* Orange line: log(|-6/2|)/l^8
* Green line: log(|-9/2|)/l^8
### Detailed Analysis
* **Blue Line (log(|-3/2|)/l^8):**
* Trend: Starts from positive infinity at x=3, decreases rapidly, and approaches 0 as x increases.
* The blue line has a vertical asymptote at x=3.
* **Orange Line (log(|-6/2|)/l^8):**
* Trend: Starts from negative infinity at x=4, increases to a maximum value around x=4.5, and then decreases, approaching 0 as x increases.
* The orange line has a vertical asymptote at x=4.
* The maximum value of the orange line is approximately 2.5e-6 at x=4.5.
* **Green Line (log(|-9/2|)/l^8):**
* Trend: Starts from negative infinity at x=4.5, increases rapidly, and approaches 0 as x increases.
* The green line has a vertical asymptote at x=4.5.
### Key Observations
* All three lines approach 0 as x increases.
* The blue line has a vertical asymptote at x=3.
* The orange line has a vertical asymptote at x=4.
* The green line has a vertical asymptote at x=4.5.
* The orange line has a maximum value around x=4.5.
### Interpretation
The chart visualizes the behavior of three logarithmic functions with different parameters. The vertical asymptotes indicate points where the functions are undefined. The trend of each line approaching 0 suggests that as x increases, the influence of the logarithmic term diminishes. The different parameters in the logarithmic functions (3/2, 6/2, 9/2) shift the position of the vertical asymptotes and affect the overall shape of the curves.
</details>
Figure 9: Entropy of $S(n,\ell)$ for $n=8$ , $k=3,6,9$ and $\ell→+∞$
The definition of clustering coefficient can be found in [3]. Let $N_{v}(t)$ be the number of edges in $G_{t}$ among neighbors of $v$ , which is the number of triangles connected to the vertex $v$ . The clustering coefficient of a graph is based on a local clustering coefficient for each vertex
$$
C[v]=\frac{N_{v}(t)}{d_{G}(v)(d_{G}(v)-1)/2},
$$
If the degree of node $v$ is $0 0$ or $1$ , then we can set $C[v]=0$ . By definition, we have $0≤ C[v]≤ 1$ for $v∈ V(G)$ .
The clustering coefficient for the whole graph $G$ is the average of the local values $C(v)$
$$
C(G)=\frac{1}{|V(G)|}\left(\sum_{v\in V(G)}C[v]\right).
$$
The clustering coefficient of a graph is closely related to the transitivity of a graph, as both measure the relative frequency of triangles [22, 24].
**Proposition 3.1**
*The clustering coefficient of generalized Sierpiński graph $S(n,\ell)$ is
$$
C(S(n,\ell))=\frac{\ell^{-n}\left(2\ell-2\ell^{n}+\ell^{n+1}\right)}{\ell}.
$$*
* Proof*
For any $v∈ V(S(n,\ell))$ , if $v$ is a extremal vertex, then $d_{S(n,\ell)}(v)=\ell-1$ and $G[\{N(v)\}]\cong K_{\ell-1}$ , and hence
$$
C[v]=\frac{N_{v}(t)}{d_{G}(v)(d_{G}(v)-1)/2}={\binom{\ell-1}{2}}/{\binom{\ell-%
1}{2}}=1.
$$
If $v$ is not a extremal vertex, then $d_{S(n,\ell)}(v)=\ell$ and $G[\{N(v)\}]\cong K_{\ell}+e$ , where $K_{\ell}+e$ is graph obtained from a complete graph $K_{\ell}$ by adding a pendent edge. Hence, we have $C[v]={\binom{\ell-1}{2}}/{\binom{\ell}{2}}=\frac{\ell-2}{\ell}$ . Since there exists $\ell$ extremal vertices in Sierpiński graph $S(n,\ell)$ , it follows that
$$
C(S(n,\ell))=\frac{1}{|V(G)|}\left(\sum_{v\in V(G)}C[v]\right)=\frac{1}{\ell^{%
n}}\left({\ell\times 1+(\ell^{n}-\ell)\frac{\ell-2}{\ell}}\right)=\frac{\ell^{%
-n}\left(2\ell-2\ell^{n}+\ell^{n+1}\right)}{\ell}
$$
∎
<details>
<summary>x6.png Details</summary>
### Visual Description
## Chart: C(S(3,l)) vs. l
### Overview
The image is a 2D line chart showing the relationship between a function C(S(3,l)) and the variable 'l'. The chart displays a curve that starts near 0.8 on the y-axis, rises sharply, and then gradually approaches a value near 0.98 as 'l' increases to 100.
### Components/Axes
* **X-axis (Horizontal):** Labeled as 'l'. The axis ranges from approximately 0 to 100, with tick marks at intervals of 20 (20, 40, 60, 80, 100).
* **Y-axis (Vertical):** Ranges from 0.8 to 1.1, with tick marks at intervals of 0.1 (0.8, 0.9, 1.0, 1.1).
* **Curve:** A blue line representing the function C(S(3,l)).
* **Label:** "C(S(3,l))" is positioned near the top-left of the chart, next to the curve's initial vertical segment.
### Detailed Analysis
* **Initial Behavior:** The curve starts at approximately l=5 and C(S(3,l)) = 0.78. It rises almost vertically to approximately C(S(3,l)) = 1.1 at l=5.
* **Rising Phase:** From l=5 to l=40, the curve rises rapidly but with decreasing slope.
* **Asymptotic Behavior:** Beyond l=40, the curve continues to rise, but at a much slower rate, approaching an asymptotic value. At l=100, C(S(3,l)) is approximately 0.98.
### Key Observations
* The function C(S(3,l)) exhibits a rapid increase for small values of 'l' and then gradually approaches a limit as 'l' increases.
* The initial vertical segment of the curve is notable.
### Interpretation
The chart illustrates a function C(S(3,l)) that is highly sensitive to small values of 'l'. The function quickly reaches a significant portion of its maximum value and then slowly converges towards a limit as 'l' becomes large. This behavior suggests that 'l' has a diminishing effect on C(S(3,l)) as 'l' increases. The function might represent a physical or mathematical relationship where the initial impact of 'l' is substantial, but its influence wanes as 'l' grows.
</details>
Figure 10: The Function of $C_{S(3,l)}$
<details>
<summary>x7.png Details</summary>
### Visual Description
## Chart: D(S(n,l)) vs l
### Overview
The image is a 2D plot showing the relationship between D(S(n,l)) and l. The x-axis represents 'l' and ranges from 2 to 16. The y-axis represents D(S(n,l)) and ranges from 0 to 30000. A single blue line represents the data, showing an exponential increase as 'l' increases.
### Components/Axes
* **X-axis:**
* Label: l
* Scale: 2 to 16, with tick marks at every integer value.
* **Y-axis:**
* Label: D(S(n,l))
* Scale: 0 to 30000, with tick marks at 5000 intervals (0, 5000, 10000, 15000, 20000, 25000, 30000).
* **Data Series:**
* Color: Blue
* Label: D(S(n,l))
### Detailed Analysis
The blue line, representing D(S(n,l)), exhibits the following behavior:
* For l values between 2 and approximately 10, the value of D(S(n,l)) remains close to 0.
* As l increases beyond 10, the value of D(S(n,l)) begins to increase gradually.
* Beyond l = 12, the increase becomes exponential, rising sharply.
* At l = 15, D(S(n,l)) is approximately 25000.
* The curve ends at approximately l = 15.5, with D(S(n,l)) near 28000.
Specific data points (approximate):
* l = 2, D(S(n,l)) ≈ 0
* l = 4, D(S(n,l)) ≈ 0
* l = 6, D(S(n,l)) ≈ 0
* l = 8, D(S(n,l)) ≈ 0
* l = 10, D(S(n,l)) ≈ 500
* l = 12, D(S(n,l)) ≈ 2000
* l = 14, D(S(n,l)) ≈ 12000
* l = 15, D(S(n,l)) ≈ 25000
### Key Observations
* The function D(S(n,l)) is relatively stable at a value near zero for small values of l.
* There is a sharp exponential increase in D(S(n,l)) as l approaches 16.
### Interpretation
The plot suggests that the function D(S(n,l)) is highly sensitive to changes in 'l' when 'l' is above a certain threshold (around 12). This could indicate a critical point or a phase transition in the underlying system being modeled by D(S(n,l)). The near-zero values for small 'l' suggest a stable or inactive state, while the exponential increase indicates a rapid change or instability as 'l' increases. The relationship between 'n' and 'l' within the function S(n,l) is not explicitly shown, but it likely plays a significant role in determining the behavior of D(S(n,l)).
</details>
Figure 11: The diameter of $S(n,\ell)$
**Theorem 3.2**
*[21] The diameter of $S(n,\ell)$ is $Diam(S(n,\ell))=2^{\ell}-1$ ;*
For network properties of Sierpiński graph $S(n,\ell)$ , the the diameter function can be seen in Figure 11 and its clustering coefficient is closely related to $1$ when $\ell\longrightarrow∞$ ; see Figure 11, which implies that the Sierpiński graph $S(n,\ell)$ is a hight transitivity graph.
4 Acknowledgments
The authors would like to thank the anonymous referee for the careful reading of the manuscript and the numerous helpful suggestions and comments.
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