The generalized 3-connectivity of Cartesian product graphs

The generalized connectivity of a graph, which was introduced recently by Chartrand et al., is a generalization of the concept of vertex connectivity. Let $S$ be a nonempty set of vertices of $G$, a collection $\{T_1,T_2,...,T_r\}$ of trees in $G$ is said to be internally disjoint trees connecting $S$ if $E(T_i)\cap E(T_j)=\emptyset$ and $V(T_i)\cap V(T_j)=S$ for any pair of distinct integers $i,j$, where $1\leq i,j\leq r$. For an integer $k$ with $2\leq k\leq n$, the $k$-connectivity $\kappa_k(G)$ of $G$ is the greatest positive integer $r$ for which $G$ contains at least $r$ internally disjoint trees connecting $S$ for any set $S$ of $k$ vertices of $G$. Obviously, $\kappa_2(G)=\kappa(G)$ is the connectivity of $G$. Sabidussi showed that $\kappa(G\Box H) \geq \kappa(G)+\kappa(H)$ for any two connected graphs $G$ and $H$. In this paper, we first study the 3-connectivity of the Cartesian product of a graph $G$ and a tree $T$, and show that $(i)$ if $\kappa_3(G)=\kappa(G)\geq 1$, then $\kappa_3(G\Box T)\geq \kappa_3(G)$; $(ii)$ if $1\leq \kappa_3(G)< \kappa(G)$, then $\kappa_3(G\Box T)\geq \kappa_3(G)+1$. Furthermore, for any two connected graphs $G$ and $H$ with $\kappa_3(G)\geq\kappa_3(H)$, if $\kappa(G)>\kappa_3(G)$, then $\kappa_3(G\Box H)\geq \kappa_3(G)+\kappa_3(H)$; if $\kappa(G)=\kappa_3(G)$, then $\kappa_3(G\Box H)\geq \kappa_3(G)+\kappa_3(H)-1$. Our result could be seen as a generalization of Sabidussi's result. Moreover, all the bounds are sharp.Comment: 17 page

Note on minimally kk-rainbow connected graphs

An edge-colored graph $G$, where adjacent edges may have the same color, is {\it rainbow connected} if every two vertices of $G$ are connected by a path whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if one can use $k$ colors to make $G$ rainbow connected. For integers $n$ and $d$ let $t(n,d)$ denote the minimum size (number of edges) in $k$-rainbow connected graphs of order $n$. Schiermeyer got some exact values and upper bounds for $t(n,d)$. However, he did not get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil$. In this paper, we improve his lower bound of $t(n,2)$, and get a lower bound of $t(n,d)$ for $3\leq d<\lceil\frac{n}{2}\rceil$.Comment: 8 page

Fault-tolerant Hamiltonian laceability of Cayley graphs generated by transposition trees

AbstractA bipartite graph is Hamiltonian laceable if there exists a Hamiltonian path joining every pair of vertices that are in different parts of the graph. It is well known that Cay(Sn,B) is Hamiltonian laceable, where Sn is the symmetric group on {1,2,…,n} and B is a generating set consisting of transpositions of Sn. In this paper, we show that for any F⊆E(Cay(Sn,B)), if |F|≤n−3 and n≥4, then there exists a Hamiltonian path in Cay(Sn,B)−F joining every pair of vertices that are in different parts of the graph. The result is optimal with respect to the number of edge faults