Two bounds for generalized $3$-connectivity of Cartesian product graphs

TL;DR

This paper establishes two new lower bounds for the generalized 3-connectivity of Cartesian product graphs, one surpassing previous bounds and the other confirming a conjecture for 3-connected graphs.

Contribution

It provides two distinct lower bounds for the generalized 3-connectivity of Cartesian product graphs, improving upon existing results and validating a conjecture.

Findings

First lower bound is stronger than previous bounds.

Second lower bound confirms the conjecture for 3-connected graphs.

Advances understanding of connectivity in product graphs.

Abstract

The generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$, which was introduced by Chartrand et al.(1984) is a generalization of the concept of vertex connectivity. Let $G$ and $H$ be nontrivial connected graphs. Recently, Li et al. gave a lower bound for the generalized $3$-connectivity of the Cartesian product graph $G \square H$ and proposed a conjecture for the case that $H$ is $3$-connected. In this paper, we give two different forms of lower bounds for the generalized $3$-connectivity of Cartesian product graphs. The first lower bound is stronger than theirs, and the second confirms their conjecture.

Equations

$${\kappa }_3(K_{a,b})=\{\begin{array}{ll}a-1& \text{if a=b},\\ a& \text{if a<b}.\end{array}$$

$$\kappa (S)\ge \{\begin{array}{ll}l+{\delta }_1-1& \text{if δ1≥t−2}\\ l+{\delta }_1-\lceil \frac{t-{\delta }_1}{2}\rceil & \text{if δ1≤t−3}.\end{array}$$

$$\kappa (S)\ge \{\begin{array}{ll}l+{\delta }_1-1& \text{if δ1≥⌊2l⌋−2},\\ l+{\delta }_1-\lceil \frac{\lfloor \frac{l}{2}\rfloor -{\delta }_1}{2}\rceil & \text{if δ1≤⌊2l⌋−3}.\end{array}$$

$${\kappa }_3(K_{a,b,c})=\{\begin{array}{ll}a+b& \text{if a+b≤c},\\ \lfloor \frac{a+b+c}{2}\rfloor & \text{if a+b≥c+1}.\end{array}$$

$${\kappa }_3(K_b\square K_{a,a,a})=\{\begin{array}{ll}2a+b-2& \text{if b≥a−1},\\ \lfloor \frac{3a+3b-3}{2}\rfloor & \text{if b≤a−2}.\end{array}$$

$${\kappa }_3(K_b\square K_{a,a+1,a+1})=\{\begin{array}{ll}2a+b-1& \text{if b≥a−1},\\ \lfloor \frac{3a+3b-1}{2}\rfloor & \text{if b≤a−2}.\end{array}$$

$$\kappa (S)\ge \{\begin{array}{ll}2a+b-1& \text{if b≥a−1},\\ \lfloor \frac{3a+3b-1}{2}\rfloor & \text{if b≤a−2}.\end{array}$$

Full text

Keywords: Connectivity, Generalized connectivity, Cartesian product

MSC 2010: 05C76 , 05C40. ††E-mail addresses: gaoh1118@yeah.net (H. Gao), bjlv@bnu.edu.cn (B. Lv),

wangks@bnu.edu.cn (K. Wang).

1 Introduction

All graphs in this paper are undirected, finite and simple. We refer to the book [1] for graph theoretic notations and terminology not described here. The generalized connectivity of a graph G, which was introduced by Chartrand et al. [2], is a natural generalization of the concept of vertex connectivity.

A tree $T$ is called an $S$-tree if $S\subseteq V(T)$. A family of $S$-trees $T_{1}$, $T_{2}$,…,$T_{r}$ are internally disjoint if $E(T_{i})\cap E(T_{j})=\phi$ and $V(T_{i})\cap V(T_{j})=S$ for any pair of integers $i$ and $j$, where $1\leq i<j\leq r$. We denote by $\kappa(S)$ the greatest number of internally disjoint $S$-trees. For an integer $k$ with $2\leq k\leq v(G)$, the generalized $k$-connectivity $\kappa_{k}(G)$ are defined to be the least value of $\kappa(S)$ when $S$ runs over all $k$-subsets of $V(G)$. Clearly, when $k=2$, $\kappa_{2}(G)=\kappa(G)$.

In addition to being a natural combinatorial notation, the generalized connectivity can be motivated by its interesting interpretation in practice. For example, suppose that $G$ represents a network. If one considers to connect a pair of vertices of $G$, then a path is used to connect them. However, if one wants to connect a set $S$ of vertices of $G$ with $|S|\geq 3$, then a tree has to be used to connect them. This kind of tree with minimum order for connecting a set of vertices is usually called Steiner tree, and popularly used in the physical design of VLSI, see [16]. Usually, one wants to consider how tough a network can be, for the connection of a set of vertices. Then, the number of totally independent ways to connect them is a measure for this purpose. The generalized $k$-connectivity can serve for measuring the capability of a network $G$ to connect any $k$ vertices in $G$.

Determining $\kappa_{k}(G)$ for most graphs is a difficult problem. In [4], Li and Li derived that for any fixed integer $k\geq 2$, given a graph $G$ and a subset $S$ of $V(G)$, deciding whether there are $k$ internally disjoint trees connecting $S$, namely deciding whether $\kappa(S)\geq k$ is NP-complete. The exact value of $\kappa_{k}(G)$ is known for only a small class of graphs. Examples are complete graphs [3], complete bipartite graphs [5], complete equipartition $3$-partite graphs [6], star graphs [15], bubble-sort graphs [15], and connected Cayley graphs on Abelian groups with small degrees [17]. Upper bounds and lower bounds of generalized connectivity of a graph have been investigated by Li et al. [9, 10, 14] and Li and Mao [12]. And Li et al. investigated Extremal problems in [8, 7]. We refer the readers to [13] for more results.

In [9], Li et al studied the generalized $3$-connectivity of Cartesian product graphs and showed the following result.

Theorem 1.1** ([9]).**

Let $G$ and $H$ be connected graphs such that $\kappa_{3}(G)\geq\kappa_{3}(H)$. The following assertions hold:

(i) if $\kappa(G)=\kappa_{3}(G)$, then $\kappa_{3}(G\square H)\geq\kappa_{3}(G)+\kappa_{3}(H)-1$. Moreover, the bound is sharp;

(ii) if $\kappa(G)>\kappa_{3}(G)$, then $\kappa_{3}(G\square H)\geq\kappa_{3}(G)+\kappa_{3}(H)$. Moreover, the bound is sharp.

Later in [11], Li et al gave a better result when $H$ becomes a $2$-connected graph.

Theorem 1.2** ([11]).**

Let $G$ be a nontrivial connected graph, and let $H$ be a 2-connected graph. The following assertions hold:

(i) if $\kappa(G)=\kappa_{3}(G)$, then $\kappa_{3}(G\square H)\geq\kappa_{3}(G)+1$. Moreover, the bound is sharp;

(ii) if $\kappa(G)>\kappa_{3}(G)$, then $\kappa_{3}(G\square H)\geq\kappa_{3}(G)+2$. Moreover, the bound is sharp.

Also in [11], Li et al proposed a conjecture as follows:

Conjecture 1.3** ([11]).**

Let $G$ be a nontrivial connected graph, and let $H$ be a $3$-connected graph. The following assertions hold:

(i) if $\kappa(G)=\kappa_{3}(G)$, then $\kappa_{3}(G\square H)\geq\kappa_{3}(G)+2$. Moreover, the bound is sharp;

(ii) if $\kappa(G)>\kappa_{3}(G)$, then $\kappa_{3}(G\square H)\geq\kappa_{3}(G)+3$. Moreover, the bound is sharp.

In this paper, we give two different forms of lower bounds for generalized $3$-connectivity of Cartesian product graphs.

Theorem 1.4**.**

Let $G$ and $H$ be nontrivial connected graphs. Then $\kappa_{3}(G\square H)\geq min\{\kappa_{3}(G)+\delta(H)$, $\kappa_{3}(H)+\delta(G)$, $\kappa(G)+\kappa(H)-1\}$.

Theorem 1.5**.**

Let $G$ be a nontrivial connected graph, and let $H$ be an $l$-connected graph. The following assertions hold:

(i) if $\kappa(G)=\kappa_{3}(G)$ and $1\leq l\leq 7$, then $\kappa_{3}(G\square H)\geq\kappa_{3}(G)+l-1$. Moreover, the bound is sharp;

(ii) if $\kappa(G)>\kappa_{3}(G)$ and $1\leq l\leq 9$, then $\kappa_{3}(G\square H)\geq\kappa_{3}(G)+l$. Moreover, the bound is sharp.

The paper is organized as follows. In Section $2$, we introduce some definitions and notations. In Section $3$, we give a proof of theorem 1.4, which induces theorem 1.1 and theorem 1.2, and confirms conjecture 1.3. In section $4$, we discuss the problem which number the connectivity of $H$ can be such that conjecture 1.3 still holds. And theorem 1.5 is our answer and there are counterexamples when $l\geq 8$ for $\kappa(G)=\kappa_{3}(G)$ and $l\geq 10$ for $\kappa(G)>\kappa_{3}(G)$.

2 Preliminaries

Let $G$ and $H$ be two graphs with $V(G)=\{u_{1},u_{2},...,u_{n}\}$ and $V(H)=\{v_{1},v_{2},...,v_{m}\}$, respectively. Let $\kappa(G)=k$, $\kappa(H)=l$, $\delta(G)=\delta_{1}$, and $\delta(H)=\delta_{2}$. And the discussion below is always based on the hypotheses.

Recall that the Cartesian product (also called the square product) of two graphs $G$ and $H$, written as $G\square H$, is the graph with vertex set $V(G)\times V(H)$, in which two vertices $(u,v)$ and $(u^{\prime},v^{\prime})$ are adjacent if and only if $u=u^{\prime}$ and $vv^{\prime}\in E(H)$, or $v=v^{\prime}$ and $uu^{\prime}\in E(G)$. By starting with a disjoint union of two graphs $G$ and $H$ and adding edges joining every vertex of $G$ to every vertex of $H$, one obtains the join of $G$ and $H$, denoted by $G\vee H$.

For any subgraph $G_{1}\subseteq G$, we use $G_{1}^{v_{j}}$ to denote the subgraph of $G\square H$ with vertex set $\{(u_{i},v_{j})|u_{i}\in V(G_{1})\}$ and edge set $\{(u_{i_{1}},v_{j})(u_{i_{2}},v_{j})|u_{i_{1}}u_{i_{2}}\in E(G_{1})\}$. Similarly, for any subgraph $H_{1}\subseteq H$, we use $H_{1}^{u_{i}}$ to denote the subgraph of $G\square H$ with vertex set $\{(u_{i},v_{j})|v_{j}\in V(H_{1})\}$ and edge set $\{(u_{i},v_{j_{1}})(u_{i},v_{j_{2}})|v_{j_{1}}v_{j_{2}}\in E(H_{1})\}$. Clearly, $G_{1}^{v_{j}}\cong G_{1}$, $H_{1}^{u_{i}}\cong H_{1}$.

Let $x\in V(G)$ and $Y\subseteq V(G)$. An $(x$, $Y)$-path is a path which starts at x, ends at a vertex of $Y$, and whose internal vertices do not belong to $Y$. A family of $k$ internally disjoint $(x$, $Y)$-paths whose terminal vertices are distinct is referred to as a $k$-fan from $x$ to $Y$.

For some $1\leq t\leq\lfloor\frac{k}{2}\rfloor$ and $s\geq t+1$, in $G$, a family $\{P_{1},P_{2},...,P_{s}\}$ of $s$ $u_{1}u_{2}-$paths is called an $(s,t)$-original-path-bundle with respect to $(u_{1},u_{2},u_{3})$, if $u_{3}$ are on $t$ paths $P_{1}$, …, $P_{t}$, and the $s$ paths have no internal vertices in common except $u_{3}$, as shown in figure 1.a. If there is not only an $(s,t)$-original-path-bundle $\{P_{1}^{{}^{\prime}},P_{2}^{{}^{\prime}},...,P_{s}^{{}^{\prime}}\}$ with respect $(u_{1},u_{2},u_{3})$, but also a family $\{M_{1},M_{2},...,M_{k-2t}\}$ of $k-2t$ internally disjoint $(u_{3},X)$-paths avoiding the vertices in $V(P_{1}^{{}^{\prime}}\cup...\cup P_{t}^{{}^{\prime}})-\{u_{1},u_{2},u_{3}\}$, where $X=V(P_{t+1}^{{}^{\prime}}\cup...\cup P_{s}^{{}^{\prime}})$, then we call the family of paths $\{P_{1}^{{}^{\prime}},P_{2}^{{}^{\prime}},...,P_{s}^{{}^{\prime}}\}\cup\{M_{1},M_{2},...,M_{k-2t}\}$ an $(s,t)$-reduced-path-bundle with respect to $(u_{1},u_{2},u_{3})$, as shown in figure 1.b.

In order to show our main results, we need the following theorems and lemmas.

Lemma 2.1**.**

([1, Fan Lemma])* Let $G$ be a $k$-connected graph, let $x$ be a vertex of $G$, and let $Y\subseteq V(G)\setminus\{x\}$. Then there exists a $k$-fan in $G$ from $x$ to $Y$.*

Theorem 2.2**.**

([1, p.219])* Let $S$ be a set of three pairwise-nonadjacent edges in a simple $3$-connected graph $G$. Then there is a cycle in $G$ containing all three edges of $S$ unless $S$ is an edge cut of $G$.*

Theorem 2.3** ([10]).**

Let $G$ be a connected graph with at least three vertices. If $G$ has two adjacent vertices with minimum degree $\delta$, then $\kappa_{3}(G)\leq\delta-1$.

Theorem 2.4** ([10]).**

Let $G$ be $k$-connected, and $u_{1},u_{2},u_{3}\in V(G)$. Then for some $0\leq t\leq\lfloor\frac{k}{2}\rfloor$, there exists a $(k,t)$-reduced-path-bundle $\{P_{1},P_{2},...,P_{k}\}\cup\{M_{1},M_{2},...,M_{k-2t}\}$ such that for any $1\leq i\leq k-2t$, the terminal vertex of $M_{i}$ is on $P_{t+i}$.

Theorem 2.5** ([10]).**

let $G$ be a connected graph with $n$ vertices. For every two integers $k$ and $r$ with $k\geq 0$ and $r\in\{0$, $1$, $2$, $3\}$, if $\kappa(G)=4k+r$, then $3k+\lceil\frac{r}{2}\rceil\leq\kappa_{3}(G)\leq\kappa(G)$. Moreover, the bound is sharp.

Lemma 2.6** ([5]).**

Let $a$, $b$ be integers such that $a+b\geq 3$ and $a\leq b$. Then,

[TABLE]

Lemma 2.7** ([11]).**

Let $C_{1}$, $C_{2}$, …, $C_{k}$ be cycles. then $\kappa_{3}(C_{1}\square C_{2}\square...\square C_{k})=2k-1$.

3 One lower bound

Lemma 3.1**.**

If $S=\{(u_{1}$, $v_{1})$, $(u_{2}$, $v_{2})$, $(u_{3}$, $v_{3})\}$, then $\kappa(S)\geq k+l-1$.

Proof.

Set $S^{{}^{\prime}}=\{(u_{i}$, $v_{j})|i$, $j=1$, $2$, $3\}$.

Case 1 $(G\square H)(S^{{}^{\prime}})$ is not isomorphic to $C_{3}\square C_{3}$.

Since $(G\square H)(S^{{}^{\prime}})$ is not isomorphic to $C_{3}\square C_{3}$, either $G(\{u_{1}$, $u_{2}$, $u_{3}\})$ or $H(\{v_{1}$, $v_{2}$, $v_{3}\})$ is not isomorphic $C_{3}$. Without loss of generality, suppose $v_{1}$ is not adjacent to $v_{2}$ in $H$. Since $H$ is $l$-connected, there exist $l$ internally disjoint paths in H from $v_{1}$ to $v_{2}$, say $P_{j}$, in which $v_{i_{j}}$ is adjacent $v_{2}$, $j=1$, $2$, …, $l$. Suppose $v_{3}$ is not in $P_{j}$, $j=1$, $2$, …, $l-1$. Also, according to Lemma 2.1, there exists an $l$-fan in $H$ from $v_{3}$ to $\{v_{i_{1}}$, $v_{i_{2}}$, …, $v_{i_{l-1}}$, $v_{1}\}$, say $Q_{j}$, which is a $v_{3}v_{i_{j}}$-path, $j=1$, $2$, …, $l-1$; and $Q_{l}$, which is a $v_{3}v_{1}$-path. Set $T_{j}=(P_{j}-v_{2})^{u_{1}}\cup G^{v_{i_{j}}}\cup(v_{2}v_{i_{j}})^{u_{2}}\cup Q_{j}^{u_{3}}$, $j=1$, $2$, …, $l-1$. It is easy notice that, for $j=1$, $2$, …, $l-1$, $(P_{j}-v_{2})^{u_{1}}$ connects $(u_{1},v_{1})$ and $(u_{1},v_{i_{j}})$; $G^{v_{i_{j}}}\cup(v_{2}v_{i_{j}})^{u_{2}}$ connects $(u_{1},v_{i_{j}})$, $(u_{2},v_{2})$ and $(u_{3},v_{i_{j}})$; and $Q_{j}^{u_{3}}$ connects $(u_{3},v_{i_{j}})$ and $(u_{3},v_{3})$. Thus, $T_{j}$ connects $(u_{1},v_{1})$, $(u_{2},v_{2})$ and $(u_{3},v_{3})$. Since $G$ is $k$-connected, there exist $k$ internally disjoint paths in $G$ from $u_{1}$ to $u_{2}$, say $R_{j}$, in which $u_{i_{j}^{{}^{\prime}}}$ is adjacent to $u_{2}$, $j=1$, $2$, …, $k$. Suppose $u_{i_{1}^{{}^{\prime}}}$, $u_{i_{2}^{{}^{\prime}}}$, …, $u_{i_{k-2}^{{}^{\prime}}}\neq u_{1}$ or $u_{3};u_{i_{k-1}^{{}^{\prime}}}\neq u_{3}$; and $u_{i_{k}^{{}^{\prime}}}\neq u_{1}$.

Case 1.1 $u_{i_{k}^{{}^{\prime}}}\neq u_{3}$

Since $G$ is $k$-connected, there exists a $k$-fan in $G$ from $u_{3}$ to $\{u_{i_{1}^{{}^{\prime}}}$, $u_{i_{2}^{{}^{\prime}}}$, …, $u_{i_{k-2}^{{}^{\prime}}}$, $u_{2}$, $u_{i_{k}^{{}^{\prime}}}\}$, say $S_{j}$, which is a $u_{3}u_{i_{j}^{{}^{\prime}}}$-path, $j=1$, $2$, …, $k-2$, $k$; and $S_{k-1}$, which is a $u_{3}u_{2}$-path. Since $H$ is $l$-connected, $H-\{v_{i_{1}}$, $v_{i_{2}}$, …, $v_{i_{l-1}}\}$ is connected. Set $T_{j}^{{}^{\prime}}=(R_{j}-u_{2})^{v_{1}}\cup(H-\{v_{i_{1}},v_{i_{2}},...,v_{i_{l-1}}\})^{u_{i_{j}^{{}^{\prime}}}}\cup(u_{i_{j}^{{}^{\prime}}}u_{2})^{v_{2}}\cup S_{j}^{v_{3}}$, $j=1$, $2$, …, $k-2$, $k$; and $T_{k-1}^{{}^{\prime}}=R_{k-1}^{v_{1}}\cup(H-\{v_{i_{1}},v_{i_{2}},...,v_{i_{l-1}}\})^{u_{2}}\cup(S_{k-1})^{v_{3}}$. It is easy to notice that, for $j=1$, $2$, …, $k-2$, $k$, $(R_{j}-u_{2})^{v_{1}}$ connects $(u_{1}$, $v_{1})$ and $(u_{i_{j}^{{}^{\prime}}}$, $v_{1})$; $(H-\{v_{i_{1}},v_{i_{2}},...,v_{i_{l-1}}\})^{u_{i_{j}^{{}^{\prime}}}}\cup(u_{i_{j}^{{}^{\prime}}}u_{2})^{v_{2}}$ connects $(u_{i_{j}^{{}^{\prime}}}$, $v_{1})$, $(u_{2}$, $v_{2})$ and $(u_{i_{j}^{{}^{\prime}}}$, $v_{3})$; and $S_{j}^{v_{3}}$ connects $((u_{i_{j}^{{}^{\prime}}}$, $v_{3}))$ and $(u_{3}$, $v_{3})$. Thus $T_{j}^{{}^{\prime}}$ connects $(u_{1}$, $v_{1})$, $(u_{2}$, $v_{2})$ and $(u_{3}$, $v_{3})$. Similarly, so does $T_{k-1}^{{}^{\prime}}$. It is clear that $T_{1}$, …, $T_{l-1}$, $T_{1}^{{}^{\prime}}$, …, $T_{k}^{{}^{\prime}}$ are pairwise disjoint except the vertex set $S$. See figure 2.

Case 1.2 $u_{i_{k}^{{}^{\prime}}}=u_{3}$

Since $G$ is $k$-connected, there exists a $(k-1)$-fan in $G-u_{1}$ from $u_{3}$ to $\{u_{i_{1}^{{}^{\prime}}}$, $u_{i_{2}^{{}^{\prime}}}$, …, $u_{i_{k-2}^{{}^{\prime}}}$, $u_{2}\}$, say $S_{j}$, which is a $u_{3}u_{i_{j}^{{}^{\prime}}}$-path, $j=1$, $2$, …, $k-2$; and $S_{k-1}$, which is a $u_{3}u_{2}$-path. Set $T_{j}^{{}^{\prime}}=(R_{j}-u_{2})^{v_{1}}\cup(H-\{v_{i_{1}},v_{i_{2}},...,v_{i_{l-1}}\})^{u_{i_{j}^{{}^{\prime}}}}\cup(u_{i_{j}^{{}^{\prime}}}u_{2})^{v_{2}}\cup S_{j}^{v_{3}}$, $j=1$, $2$, …, $k-2$; $T_{k-1}^{{}^{\prime}}=R_{k-1}^{v_{1}}\cup(H-\{v_{i_{1}}$, $v_{i_{2}}$, …, $v_{i_{l-1}}\})^{u_{2}}\cup S_{k-1}^{v_{3}}$; and $T_{k}^{{}^{\prime}}=Q_{l}^{u_{3}}\cup(R_{k}-u_{2})^{v_{1}}\cup P_{l}^{u_{1}}\cup R_{k-1}^{v_{2}}$. It is easy to notice that $Q_{l}^{u_{3}}$ connects $(u_{3}$, $v_{3})$ and $(u_{3}$, $v_{1})$; $(R_{k}-u_{2})^{v_{1}}$ connects $(u_{3}$, $v_{1})$ and $(u_{1}$, $v_{1})$; $P_{l}^{u_{1}}$ connects $(u_{1}$, $v_{1})$ and $(u_{1}$, $v_{2})$; and $R_{k-1}^{v_{2}}$ connects $(u_{1}$, $v_{2})$ and $(u_{2}$, $v_{2})$. Thus, $T_{k}^{{}^{\prime}}$ connects $(u_{1}$, $v_{1})$, $(u_{2}$, $v_{2})$ and $(u_{3}$, $v_{3})$. It is clear that $T_{1}$, …, $T_{l-1}$, $T_{1}^{{}^{\prime}}$, …, $T_{k}^{{}^{\prime}}$ are pairwise disjoint except the vertex set $S$. See figure 3.

Case 2 $(G\square H)(S^{{}^{\prime}})\cong C_{3}\square C_{3}$.

Because of lemma 2.7, there exist $3$ internally disjoint $S$-trees in $(G\square H)(S^{{}^{\prime}})$, say $T_{j}^{{}^{\prime\prime}}$, $j=1$, $2$, $3$. Since $H$ is $l$-connected, $\kappa$ $((H-v_{3})-v_{1}v_{2})\geq l-2$. Thus there exist $l-2$ internally disjoint $v_{1}v_{2}$-paths in $(H-v_{3})-v_{1}v_{2}$, say $P_{j}$, in which $v_{i_{j}}$ is adjacent of $v_{2}$, $j=1$, $2$, …, $l-2$. By Lemma 2.1, there exists an $(l-2)$-fan in $H-v_{1}-v_{2}$ from $v_{3}$ to $\{v_{i_{1}}$, $v_{i_{2}}$, …, $v_{i_{l-2}}\}$. Similarly to Case 1, we can construct $l-2$ internally disjoint $S$-trees $T_{1}$, $T_{2}$, …, $T_{l-2}$. Also, because $\kappa(G)=k$, another $k-2$ $S$-trees $T_{1}^{{}^{\prime}}$, $T_{2}^{{}^{\prime}}$, …, $T_{k-2}^{{}^{\prime}}$ can be constructed. It is clear that $T_{1}$, …, $T_{l-2}$, $T_{1}^{{}^{\prime}}$, …, $T_{k-2}^{{}^{\prime}}$, $T_{1}^{{}^{\prime\prime}}$, $T_{2}^{{}^{\prime\prime}}$, $T_{3}^{{}^{\prime\prime}}$ are internally disjoint $S$-trees. ∎

Lemma 3.2**.**

If $S=\{(u_{1}$, $v_{1})$, $(u_{1}$, $v_{2})$, $(u_{2}$, $v_{1})\}$, then $\kappa(S)\geq k+l-1$.

Proof.

Since $H$ is $l$-connected, there exist $l$ internally disjoint $v_{1}v_{2}$-paths in $H$, say $P_{j}$, in which $v_{i_{j}}$ is adjacent to $v_{1}$, $j=1$, $2$, …, $l$. Suppose $v_{i_{j}}\neq v_{2}$, $j=1$, $2$, …, $l-1$. Set $T_{j}=P_{j}^{u_{1}}\cup G^{v_{i_{j}}}\cup(v_{1}v_{i_{j}})^{u_{2}}$, $j=1$, $2$, …, $l-1$. Since $G$ is $k$-connected, there exist $k$ internally disjoint $u_{1}u_{2}$-paths in $G$, say $Q_{j}$, in which $u_{i_{j}^{{}^{\prime}}}$ is adjacent to $u_{1}$, $j=1$, $2$, …, $k$. Suppose $u_{i_{j}^{{}^{\prime}}}\neq u_{2}$, $j=1$, $2$, …, $k-1$. Set $T_{j}^{{}^{\prime}}=Q_{j}^{v_{1}}\cup(H-\{v_{i_{1}},v_{i_{2}},...,v_{i_{l-1}}\})^{u_{i_{j}^{{}^{\prime}}}}\cup(u_{1}u_{i_{j}^{{}^{\prime}}})^{v_{2}}$, $j=1$, $2$, …, $k-1$; and $T^{{}^{\prime\prime}}=P_{l}^{u_{1}}\cup Q_{k}^{v_{1}}$. It is clear that $T_{1}$, …, $T_{l-1}$, $T_{1}^{{}^{\prime}}$, …, $T_{k-1}^{{}^{\prime}}$, $T^{{}^{\prime\prime}}$ are connected graphs containing $S$ and are pairwise disjoint except $S$. See figure 4. ∎

Lemma 3.3**.**

If $S=\{(u_{1}$, $v_{1})$, $(u_{2}$, $v_{1})$, $(u_{3}$, $v_{2})\}$, then $\kappa(S)\geq k+l-1$.

Proof.

Since $H$ is $l$-connected, there exist $l$ internally disjoint $v_{1}v_{2}$-paths in $H$, say $P_{j}$, in which $v_{i_{j}}$ is adjacent to $v_{1}$, $j=1$, $2$, …, $l$. Suppose $v_{i_{j}}\neq v_{2}$, $j=1$, $2$, …, $l-1$. Set $T_{j}=(v_{1}v_{i_{j}})^{u_{1}}\cup(v_{1}v_{i_{j}})^{u_{2}}\cup G^{v_{i_{j}}}\cup(P_{j}-v_{1})^{u_{3}}$, $j=1$, $2$, …, $l-1$. Since $G$ is $k$-connected, there exist $k$ internally disjoint $u_{1}u_{2}$-paths in $G$, say $Q_{j}$, in which $u_{i_{j}^{{}^{\prime}}}$ is adjacent $u_{1}$, $j=1$, $2$, …, $k$. Suppose $u_{i_{j}^{{}^{\prime}}}\neq u_{2}$, $u_{3}$, $j=1$, $2$, …, $k-2$. Due to Lemma 2.1, there exists a $k$-fan in $G$ from $u_{3}$ to $\{u_{i_{1}^{{}^{\prime}}}$, $u_{i_{2}^{{}^{\prime}}}$, …, $u_{i_{k-2}^{{}^{\prime}}}$, $u_{1}$, $u_{2}\}$, say $R_{j}$, which is a $u_{3}u_{i_{j}^{{}^{\prime}}}$-path, $j=1$, $2$, …, $k-2$; $R_{k-1}$, which is a $u_{3}u_{1}$-path; and $R_{k}$, which is a $u_{3}u_{2}$-path. Set $T_{j}^{{}^{\prime}}=Q_{j}^{v_{1}}\cup(H-\{v_{i_{1}}$, $v_{i_{2}}$, …, $v_{i_{l-1}}\})^{u_{i_{j}^{{}^{\prime}}}}\cup R_{j}^{v_{2}}$, $j=1$, $2$, …, $k-2$; $T_{k-1}^{{}^{\prime}}=Q_{k-1}^{v_{1}}\cup(H-\{v_{i_{1}}$, $v_{i_{2}}$, …, $v_{i_{l-1}}\})^{u_{1}}\cup R_{k-1}^{v_{2}}$; and $T_{k}^{{}^{\prime}}=Q_{k}^{v_{1}}\cup(H-\{v_{i_{1}}$, $v_{i_{2}}$, …, $v_{i_{l-1}}\})^{u_{2}}\cup R_{k}^{v_{2}}$. It is clear that $T_{1}$, …, $T_{l-1}$, $T_{1}^{{}^{\prime}}$, …, $T_{k}^{{}^{\prime}}$ are connected graphs containing $S$ and are pairwise disjoint except $S$. See figure 5. ∎

Lemma 3.4**.**

If $S=\{(u_{1}$, $v_{1})$, $(u_{2}$, $v_{1})$, $(u_{3}$, $v_{1})\}$, then $\kappa(S)\geq\kappa_{3}(G)+\delta(H)$.

Proof.

Let $T_{1}$, $T_{2}$, …, $T_{\kappa_{3}(G)}$ be the internally disjoint $\{u_{1}$, $u_{2}$, $u_{3}\}$-trees in $G$. Let $v_{i_{1}}$, $v_{i_{2}}$, …, $v_{i_{\delta(H)}}$ be neighbors of $v_{1}$ in $H$. Set $T_{j}^{{}^{\prime}}=(v_{1}v_{i_{j}})^{u_{1}}\cup(v_{1}v_{i_{j}})^{u_{2}}\cup(v_{1}v_{i_{j}})^{u_{3}}\cup G^{v_{i_{j}}}$, $j=1$, $2$, …, $\delta(H)$. It is clear that $T_{1}^{v_{1}}$, $T_{2}^{v_{1}}$, …, $T_{\kappa_{3}(G)}^{v_{1}}$, $T_{1}^{{}^{\prime}}$, $T_{2}^{{}^{\prime}}$, …, $T_{\delta(H)}^{{}^{\prime}}$ are connected graphs containing $S$ and are pairwise disjoint except $S$. ∎

From Lemma 3.1 to Lemma 3.4, without loss of generality, we discuss all positions of three vertices of $G\square H$. Hence, theorem 1.4 is obvious.

Theorem 1.4. Let $G$ and $H$ be nontrivial connected graphs. Then $\kappa_{3}(G\square H)\geq min\{\kappa_{3}(G)+\delta(H)$, $\kappa_{3}(H)+\delta(G)$, $\kappa(G)+\kappa(H)-1\}$.

Example 3.5**.**

Let $a$ and $b$ be integers such that $a\geq 1$ and $b\geq 2$. Then $\kappa_{3}(K_{a+1}\square K_{b})=\kappa_{3}((K_{a}\vee\overline{K_{2}})\square K_{b})=a+b-2$.

Proof.

According to theorem 2.3, $\kappa_{3}(K_{a+1}\square K_{b})$, $\kappa_{3}((K_{a}\vee\overline{K_{2}})\square K_{b})\leq a+b-2$. It is easy to see that $\kappa(K_{b})=b-1$; $\kappa_{3}(K_{b})=b-2$, if $b\geq 3$; and $\kappa(K_{a}\vee\overline{K_{2}})=\kappa_{3}(K_{a}\vee\overline{K_{2}})=a$. From lemma 3.1-3.4, we can see that $\kappa_{3}(K_{a+1}\square K_{b})$, $\kappa_{3}((K_{a}\vee\overline{K_{2}})\square K_{b})\geq a+b-2$. ∎

Due to theorem 2.4, $\delta(G)\geq\kappa(G)\geq\kappa_{3}(G)$, $\delta(H)\geq\kappa(H)\geq\kappa_{3}(H)$. Hence, theorem 1.4 induces theorem 1.1.

Corollary 3.6**.**

Let $G$ be a nontrivial connected graph, and $H$ an $l$-connected graph, where $1\leq l\leq 5$. The following assertions hold;

(i) if $\kappa(G)=\kappa_{3}(G)$, then $\kappa_{3}(G\square H)\geq\kappa_{3}(G)+l-1$. Moreover, the bound is sharp;

(ii) if $\kappa(G)>\kappa_{3}(G)$, then $\kappa_{3}(G\square H)\geq\kappa_{3}(G)+l$. Moreover, the bound is sharp.

Proof.

Due to theorem 2.4, $\kappa_{3}(H)\geq l-1$. Hence, $\kappa_{3}(H)+\delta(G)\geq\kappa(G)+l-1$. And theorem 1.4 induces this corollary. Example 3.5 guarantees the sharpness. ∎

Obviously, corollary 3.6 induces theorem 1.2 and confirms conjecture 1.3. However, corollary 3.6 does not answer the question whether conjecture 1.3 still holds if $H$ is $l$-connected for $l\geq 6$. And this is what will discuss in the next section.

4 Another lower bound

Lemma 4.1**.**

Let $0\leq t\leq\lfloor\frac{l}{2}\rfloor$, and $S=\{(u_{1},v_{1}),(u_{1},v_{2}),(u_{1},v_{3})\}$. If there exists an $(l,t)$-reduced-path-bundle $\{P_{1},P_{2},...,P_{l}\}\cup\{M_{1},M_{2},...,M_{l-2t}\}$ in $H$ such that for any $1\leq i\leq l-2t$, the terminal vertex of $M_{i}$ is on $P_{t+i}$, then

[TABLE]

Proof.

For any $j\in\{1,2,...,t\}$, $P_{j}$ is divided into two paths by $v_{3}$, denoted by $P_{j1}$ and $P_{j2}$, where $P_{j1}$ is a $v_{1}v_{3}$-path and $P_{j2}$ is a $v_{3}v_{2}$-path. Without loss of generality, suppose $u_{1}u_{2}$, $u_{1}u_{3}$, …, $u_{1}u_{\delta_{1}+1}\in E(G)$; $l(P_{j1})$, $l(P_{j2})$, $l(P_{l-j+1})\geq 2$, $j=1$, $2$, …, $t-1$. Set $T_{j}=(M_{j}\cup P_{t+j})^{u_{1}}$, $j=1$, $2$, …, $l-2t$.

Case 1 $t\leq 2$.

Set $T_{j}^{{}^{\prime}}=(u_{1}u_{j+1})^{v_{1}}\cup(u_{1}u_{j+1})^{v_{2}}\cup(u_{1}u_{j+1})^{v_{3}}\cup H^{u_{j+1}}$, $j=1$, $2$, …, $\delta_{1}$. If $t=0$, then $\kappa(S)\geq l-2t+\delta_{1}=l+\delta_{1}$. If $t=1$, then set $T^{{}^{\prime\prime}}=P_{1}^{u_{1}}$. So $\kappa(S)\geq l-2t+\delta_{1}+1=l+\delta_{1}-1$. If $t=2$, then set $T_{1}^{{}^{\prime\prime}}=(P_{l}\cup P_{11})^{u_{1}}$, $T_{2}^{{}^{\prime\prime}}=(P_{21}\cup P_{12})^{u_{1}}$, and $T_{3}^{{}^{\prime\prime}}=(P_{22}\cup P_{l-1})^{u_{1}}$. So $\kappa(S)\geq l-2t+\delta_{1}+3=l+\delta_{1}-1$.

Case 2 $t\geq 3$.

Since $t\leq\lfloor\frac{l}{2}\rfloor$, $l\geq 6$. Suppose $v_{1}v_{i_{1}}$, $v_{1}v_{i_{2}}$, $v_{3}v_{i_{3}}$, $v_{3}v_{i_{4}}$, $v_{2}v_{i_{5}}$, $v_{2}v_{i_{6}}\in E(G)$, as shown in figure 6. It is easy to see that $\kappa((H-\{v_{1},v_{2},v_{3}\})\cup\{v_{i_{1}}v_{i_{2}},v_{i_{3}}v_{i_{4}},v_{i_{5}}v_{i_{6}}\})\geq l-3\geq 3$. If $\{v_{i_{1}}v_{i_{2}},v_{i_{3}}v_{i_{4}},v_{i_{5}}v_{i_{6}}\}$ is not an edge cut of $(H-\{v_{1},v_{2},v_{3}\})\cup\{v_{i_{1}}v_{i_{2}},v_{i_{3}}v_{i_{4}},v_{i_{5}}v_{i_{6}}\}$, then according to lemma 2.2, there is a cycle containing $\{v_{i_{1}}v_{i_{2}},v_{i_{3}}v_{i_{4}},v_{i_{5}}v_{i_{6}}\}$ in $(H-\{v_{1},v_{2},v_{3}\})\cup\{v_{i_{1}}v_{i_{2}},v_{i_{3}}v_{i_{4}},v_{i_{5}}v_{i_{6}}\}$, which without loss of generality, we suppose to be $v_{i_{1}}v_{i_{2}}C_{1}v_{i_{3}}v_{i_{4}}C_{2}v_{i_{5}}v_{i_{6}}C_{3}v_{i_{1}}$, where $C_{1}$, $C_{2}$, $C_{3}$ are paths. Set $T_{1}^{{}^{\prime}}=P_{12}^{u_{1}}\cup(u_{1}u_{2})^{v_{1}}\cup(u_{1}u_{2})^{v_{i_{3}}}\cup(v_{1}v_{i_{2}})^{u_{2}}\cup C_{1}^{u_{2}}$, $T_{2}^{{}^{\prime}}=P_{l}^{u_{1}}\cup(u_{1}u_{2})^{v_{3}}\cup(u_{1}u_{2})^{v_{i_{5}}}\cup(v_{3}v_{i_{4}})^{u_{2}}\cup C_{2}^{u_{2}}$, and $T_{3}^{{}^{\prime}}=P_{11}^{u_{1}}\cup(u_{1}u_{2})^{v_{2}}\cup(u_{1}u_{2})^{v_{i_{1}}}\cup(v_{2}v_{i_{6}})^{u_{2}}\cup C_{3}^{u_{2}}$. Hence, we can find $3$ internally disjoint $S$-trees in $(\underset{v\in V(P_{1}\cup P_{l})}{\cup}(u_{1}u_{2})^{v})\cup(P_{1}\cup P_{l})^{u_{1}}\cup H^{u_{2}}$.

Case 2.1 $l\geq 7$.

Since $\kappa((H-\{v_{1},v_{2},v_{3}\})\cup\{v_{i_{1}}v_{i_{2}},v_{i_{3}}v_{i_{4}},v_{i_{5}}v_{i_{6}}\})\geq l-3\geq 4$, $\{v_{i_{1}}v_{i_{2}},v_{i_{3}}v_{i_{4}},v_{i_{5}}v_{i_{6}}\}$ cannot be an edge cut of $(H-\{v_{1},v_{2},v_{3}\})\cup\{v_{i_{1}}v_{i_{2}},v_{i_{3}}v_{i_{4}},v_{i_{5}}v_{i_{6}}\}$.

Similar to the discussion above, if $\delta_{1}\geq t-2$, without loss of generality, suppose we can find $3$ internally disjoint $S$-trees in $(\underset{v\in V(P_{j}\cup P_{l-j+1})}{\cup}(u_{1}u_{1+j})^{v})\cup(P_{j}\cup P_{l-j+1})^{u_{1}}\cup H^{u_{1+j}}$ respectively, $j=1$, $2$, …, $t-2$. Set $T_{j}^{{}^{\prime\prime}}=(u_{1}u_{j})^{v_{1}}\cup(u_{1}u_{j})^{v_{2}}\cup(u_{1}u_{j})^{v_{3}}\cup H^{u_{j}}$, $j=t$, $t+1$, …, $\delta_{1}+1$; $T_{1}^{{}^{\prime\prime\prime}}=(P_{l-t+2}\cup P_{t-1,1})^{u_{1}}$; $T_{2}^{{}^{\prime\prime\prime}}=(P_{t1}\cup P_{t-1,2})^{u_{1}}$; and $T_{3}^{{}^{\prime\prime\prime}}=(P_{t2}\cup P_{l-t+1})^{u_{1}}$. Hence, $\kappa(S)\geq l-2t+3(t-2)+\delta_{1}-t+2+3=l+\delta_{1}-1$.

If $\delta_{1}\leq t-3$, without loss of generality, suppose we can find $3$ internally disjoint $S$-trees in $(\underset{v\in V(P_{j}\cup P_{l-j+1})}{\cup}(u_{1}u_{1+j})^{v})\cup(P_{j}\cup P_{l-j+1})^{u_{1}}\cup H^{u_{1+j}}$ respectively, $j=1$, $2$, …, $\delta_{1}$. Clearly, there exist at least $\lfloor\frac{3(t-\delta_{1})}{2}\rfloor$ internally disjoint $S$-trees in $(\overset{t}{\underset{j=\delta_{1}+1}{\cup}}(P_{j1}\cup P_{j2}\cup P_{l-j+1}))^{u_{1}}$. Hence, $\kappa(S)\geq l-2t+3\delta_{1}+\lfloor\frac{3(t-\delta_{1})}{2}\rfloor=l+\delta_{1}-\lceil\frac{t-\delta_{1}}{2}\rceil$.

Case 2.2 $l=6$.

Since $3\leq t\leq\lfloor\frac{l}{2}\rfloor$, $t=3$. If $\{v_{i_{1}}v_{i_{2}},v_{i_{3}}v_{i_{4}},v_{i_{5}}v_{i_{6}}\}$ is not an edge cut of $(H-\{v_{1},v_{2},v_{3}\})\cup\{v_{i_{1}}v_{i_{2}},v_{i_{3}}v_{i_{4}},v_{i_{5}}v_{i_{6}}\}$, then similar to the case that $l\geq 7$ and $\delta_{1}\geq t-2$, $\kappa(S)\geq 5+\delta_{1}$. From now on, suppose the contrary, that is $\kappa(S)\leq 4+\delta_{1}$. Then, $u_{i_{1}}u_{i_{2}}$, $u_{i_{3}}u_{i_{4}}$, $u_{i_{5}}u_{i_{6}}\in E(H)$; and $\{v_{i_{1}}v_{i_{2}},v_{i_{3}}v_{i_{4}},v_{i_{5}}v_{i_{6}}\}$ separates $H-\{v_{1},v_{2},v_{3}\}$ into two components $C_{1}$ and $C_{2}$, where we suppose $v_{i_{1}},v_{i_{3}},v_{i_{5}}\in V(C_{1})$, and $v_{i_{2}},v_{i_{4}},v_{i_{6}}\in V(C_{2})$. Suppose $l(P_{31})\geq 2$, and $v_{i_{7}}$ is adjacent to $v_{1}$ in $P_{31}$. Since $\kappa(S)\leq 4+\delta_{1}$, so $v_{i_{1}}v_{i_{7}}$, $v_{i_{2}}v_{i_{7}}\in E(G)$, which induces that $H-\{v_{1},v_{2},v_{3}\}-\{v_{i_{1}}v_{i_{2}},v_{i_{3}}v_{i_{4}},v_{i_{5}}v_{i_{6}}\}$ is connected, a contradiction. Hence, $l(P_{31})=1$. Similarly, $l(P_{32})=l(P_{4})=1$. Suppose $l(P_{11})\geq 3$, and $v_{i_{7}}$ and $v_{i_{8}}$ are adjacent to $v_{3}$ in $P_{11}$ and $P_{21}$ respectively. Since $\kappa(S)\leq 4+\delta_{1}$, $v_{i_{7}}v_{i_{8}}\in E(H)$, which induces that $H-\{v_{1},v_{2},v_{3}\}-\{v_{i_{1}}v_{i_{2}},v_{i_{3}}v_{i_{4}},v_{i_{5}}v_{i_{6}}\}$ is connected, a contradiction. Hence, $l(P_{11})=2$. Similarly, $l(P_{12})=l(P_{21})=l(P_{22})=l(P_{5})=l(P_{6})=2$. Suppose $v_{i_{1}}v_{2}\in E(G)$. Then in $H$, $\kappa(\{v_{1},v_{2},v_{3}\})\geq 5$, and according to lemma 3.4, $\kappa(S)\geq 5+\delta_{1}$, a contradiction. Hence, $v_{i_{1}}v_{2}\notin E(G)$. Since $d(v_{i_{1}})\geq 6$, there exists a vertex $v_{i_{9}}\in V(C_{1})-\{v_{i_{1}},v_{i_{3}},v_{i_{5}}\}$ such that $v_{i_{9}}v_{i_{1}}\in E(H)$. Since $\kappa(H)=6$, so there exists a $6$-fan in $H$ from $v_{i_{9}}$ to $\{v_{1},v_{2},v_{3},v_{i_{1}},v_{i_{3}},v_{i_{5}}\}$, disjoint from $C_{2}$. Then in $H$, $\kappa(\{v_{1},v_{2},v_{3}\})\geq 5$, also a contradiction. ∎

Proposition 4.2**.**

Let $S=\{(u_{1},v_{1}),(u_{1},v_{2}),(u_{1},v_{3})\}$. Then

[TABLE]

Proof.

Since $H$ is $l$-connected, so according to theorem 2.4, for some $0\leq t\leq\lfloor\frac{l}{2}\rfloor$, there exists an $(l,t)$-reduced-path-bundle $\{P_{1},P_{2},...,P_{l}\}\cup\{M_{1},M_{2},...,M_{l-2t}\}$ such that for any $1\leq i\leq l-2t$, the terminal vertex of $M_{i}$ is on $P_{t+i}$. When $\delta_{1}\geq\lfloor\frac{l}{2}\rfloor-2$, Since $t\leq\lfloor\frac{l}{2}\rfloor$, $\delta_{1}\geq t-2$. Hence, due to lemma 4.1, $\kappa(S)\geq l+\delta_{1}-1$. When $\delta_{1}\leq\lfloor\frac{l}{2}\rfloor-3$, if $\delta_{1}\geq t-2$, then $\kappa(S)\geq l+\delta_{1}-1\geq l+\delta_{1}-\lceil\frac{\lfloor\frac{l}{2}\rfloor-\delta_{1}}{2}\rceil$; if $\delta_{1}\leq t-3$, according to lemma 4.1, $\kappa(S)\geq l+\delta_{1}-\lceil\frac{t-\delta_{1}}{2}\rceil\geq l+\delta_{1}-\lceil\frac{\lfloor\frac{l}{2}\rfloor-\delta_{1}}{2}\rceil$. ∎

Theorem 1.5 Let $G$ be a nontrivial connected graph, and let $H$ be an $l$-connected graph. The following assertions hold:

(i) if $\kappa(G)=\kappa_{3}(G)$ and $1\leq l\leq 7$, then $\kappa_{3}(G\square H)\geq\kappa_{3}(G)+l-1$. Moreover, the bound is sharp;

(ii) if $\kappa(G)>\kappa_{3}(G)$ and $1\leq l\leq 9$, then $\kappa_{3}(G\square H)\geq\kappa_{3}(G)+l$. Moreover, the bound is sharp.

Proof.

Let $S$ be a $3$-vertex-set of $V(G\square H)$. If there does not exist a vertex $u\in V(G)$ such that $S\subseteq V(H^{u})$, then according to lemma 3.1-lemma 3.4, $\kappa(S)\geq\kappa_{3}(G)+\delta(H)\geq\kappa_{3}(G)+l$ or $\kappa(S)\geq\kappa(G)+l-1$. Next, without loss of generality, suppose $S=\{(u_{1},v_{1}),(u_{1},v_{2}),(u_{1},v_{3})\}$. If $l\leq 7$, then $\delta_{1}\geq\lfloor\frac{l}{2}\rfloor-2$, which according to proposition 4.2, induces that $\kappa(S)\geq l+\delta_{1}-1$. If $l=8$ or $9$ and $\kappa(G)>\kappa_{3}(G)$, then $\delta_{1}\geq 2=\lfloor\frac{l}{2}\rfloor-2$, which according to proposition 4.2, induces that $\kappa(S)\geq l+\delta_{1}-1\geq\kappa_{3}(G)+l$. Example 3.5 guarantees the sharpness. ∎

An $S$-tree $T$ in $G$ is said to be minimal if for any $S$-tree $T^{{}^{\prime}}$, $V(T^{{}^{\prime}})\subseteq V(T)$ and $\partial{(S)}\cap E(T^{{}^{\prime}})\subseteq\partial{(S)}\cap E(T)$ induce $V(T^{{}^{\prime}})=V(T)$ and $\partial{(S)}\cap E(T^{{}^{\prime}})=\partial{(S)}\cap E(T)$, where $\partial(S)$ denotes the set of edges with one vertex in $S$.

Example 4.3**.**

Let $a$, $b$, $c$ be integers such that $1\leq a\leq b\leq c$. Then,

[TABLE]

Proof.

It is easy to see that $\kappa_{3}(K_{1,1,1})=1$ and $\kappa_{3}(K_{1,1,c})=2$ for $c\geq 2$. Next, suppose $a+b\geq 3$. Let $V_{1}=\{v_{11},v_{21},...,v_{a1}\}$, $V_{2}=\{v_{12},v_{22},...,v_{b2}\}$ and $V_{3}=\{v_{13},v_{23},...,v_{c3}\}$ be the maximal independent sets of $K_{a,b,c}$. Let $S$ be a $3$-vertex-set of $V(K_{a,b,c})$. If $S\subseteq V_{1}\cup V_{2}$, then $\kappa(S)\geq c+\kappa_{3}(K_{a,b})$; If $S\subseteq V_{1}\cup V_{3}$, then $\kappa(S)\geq b+\kappa_{3}(K_{a,c})$; If $S\subseteq V_{2}\cup V_{3}$, then $\kappa(S)\geq a+\kappa_{3}(K_{b,c})$. Moreover, equalities can hold in the three inequalities above. Next, without loss of generality, suppose $S=\{v_{11},v_{12},v_{13}\}$. Then figure 7 shows three types of minimal $S$-trees, namely A, B and C. Let $\mathscr{T}$ be the set of $\kappa(S)$ internally disjoint minimal $S$-trees, containing $u$ trees of type A (or A trees), $v$ trees of type B (or B trees) and $w$ trees of type C (or C trees). We notice that $v+2w\leq 3$ and $2u+v\leq a+b+c-3$. Hence, $\kappa(S)=u+v+w\leq\lfloor\frac{a+b+c}{2}\rfloor$. Besides, $\kappa(S)\leq d(v_{13})=a+b$. Set $T_{1}=(v_{11}v_{12})\cup(v_{12}v_{13})$ and $T_{2}=(v_{11}v_{13})\cup(v_{11}v_{23})\cup(v_{12}v_{23})$. If $a+b\geq c+1$, then it is easy to see that there exist $\lfloor\frac{a-1+b-1+c-2}{2}\rfloor=\lfloor\frac{a+b+c}{2}\rfloor-2$ internally disjoint A trees in $K_{a,b,c}-v_{23}-\{v_{11}v_{12},v_{11}v_{13},v_{12}v_{13}\}$. So $\kappa(S)=\lfloor\frac{a+b+c}{2}\rfloor$. If $a+b\leq c$, then it is easy to see that there exist $a-1+b-1=a+b-2$ internally disjoint A trees in $K_{a,b,c}-v_{23}-\{v_{11}v_{12},v_{11}v_{13},v_{12}v_{13}\}$. So $\kappa(S)=a+b$. Finally, because of lemma 2.6, the problem is solved by comparing sizes. ∎

Example 4.4**.**

Let $a$ and $b$ be integers such that $a\geq 1$ and $b\geq 2$. Then,

[TABLE]

and

[TABLE]

Proof.

Since the proofs are quite similar for $K_{b}\square K_{a,a,a}$ and $K_{b}\square K_{a,a+1,a+1}$, we give our proof only for the latter. Let $V(K_{b})=\{u_{1},u_{2},...,u_{b}\}$; Let $V_{1}=\{v_{11},v_{21},...,v_{a1}\}$, $V_{2}=\{v_{12},v_{22},...,v_{a+1,2}\}$ and $V_{3}=\{v_{13},v_{23},...,v_{a+1,3}\}$ be the maximal independent sets of $K_{a,a+1,a+1}$; and Let $S$ be a $3$-vertex-set of $V(K_{b}\square K_{a,a+1,a+1})$. Because of theorem 2.3, $\kappa_{3}(K_{b}\square K_{a,a+1,a+1})\leq 2a+b-1$. If for any $u\in V(G)$, $S\nsubseteq V(H^{u})$, then due to lemma 3.1-3.4, $\kappa(S)\geq 2a+b-1$; If there exists a vertex $u\in V(G)$ such that for some $i\in\{1,2,3\}$, $S\subseteq(V(H)-V_{i})^{u}$, then according to the proof of lemma 3.4, $\kappa(S)\geq 2a+b-1$. Next, without loss of generality, let $S=\{(u_{1},v_{11}),(u_{1},v_{12}),(u_{1},v_{13})\}$. Then according to proposition 4.2, we have

[TABLE]

Hence, when $b\geq a-1$, $\kappa_{3}(K_{b}\square K_{a,a+1,a+1})=2a+b-1$. Next, we claim that when $b\leq a-2$, $\kappa(S)\leq\lfloor\frac{3a+3b-1}{2}\rfloor$.

Let $F$ be the graph obtained from $K_{b}\square K_{a,a+1,a+1}$ by joining every pair of nonadjacent vertices in $\{(u,v)|u\in V(K_{b})-u_{1},v\in V(K_{a,a+1,a+1})\}$. Obviously, the maximum number of internally disjoint $S$-trees in $F$ is not less than the maximum number of internally disjoint $S$-trees in $K_{b}\square K_{a,a+1,a+1}$. In $F$, all types of minimal $S$-trees are shown in figure 7 and figure 8. Let $\mathscr{T}$ be the set with maximum number of internally disjoint minimal $S$-trees in F and as many F trees as possible. Let the number of A, B, C, D, E and F trees $\mathscr{T}$ contains be $u$, $v$, $w$, $x$, $y$ and $z$, denoted by $\mathscr{T}=uA+vB+wC+xD+yE+zF$. It is easy to see that one A tree and one D tree can be replaced by two F trees, denoted by $A+D\rightarrow 2F$. So A trees and D trees can not be both in $\mathscr{T}$. Similarly, we have $A+E\rightarrow B+F$. If $x>0$ or $y>0$, then $u=0$. We notice that $3x+2y+z\leq 3(b-1)$ and $v+2w+y\leq 3$, so $|\mathscr{T}|=v+w+x+y+z\leq 3b<\lfloor\frac{3a+3b-1}{2}\rfloor$. If $x=y=0$, then $z=3(b-1)$ and $|\mathscr{T}|=3(b-1)+\kappa_{3}(K_{a-b+1,a-b+2,a-b+2})=\lfloor\frac{3a+3b-1}{2}\rfloor$. ∎

Also, we can similarly prove that when $a\geq 4$, $\kappa_{3}(P_{3}\square K_{a,a,a})=\kappa_{3}(K_{2}\square K_{a,a,a})=\lfloor\frac{3a+3}{2}\rfloor$, and $\kappa_{3}(P_{3}\square K_{a,a+1,a+1})=\kappa_{3}(K_{2}\square K_{a,a+1,a+1})=\lfloor\frac{3a+5}{2}\rfloor$. In theorem 1.5, let $G=P_{3}$, and $H=K_{a,a,a}$ or $K_{a,a+1,a+1}$. Then $l\geq 8$, and $\kappa_{3}(G\square H)<\kappa_{3}(G)+l-1$. When $a\geq 5$, $\kappa_{3}(K_{3}\square K_{a,a,a})=\lfloor\frac{3a+6}{2}\rfloor$, and $\kappa_{3}(K_{3}\square K_{a,a+1,a+1})=\lfloor\frac{3a+8}{2}\rfloor$. In theorem 1.5, let $G=K_{3}$, and $H=K_{a,a,a}$ or $K_{a,a+1,a+1}$. Then $l\geq 10$, and $\kappa_{3}(G\square H)<\kappa_{3}(G)+l$.