On the super connectivity of Kronecker products of graphs

In this paper we present the super connectivity of Kronecker product of a general graph and a complete graph.Comment: 8 page

Polytopality and Cartesian products of graphs

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we provide several families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.Comment: 21 pages, 10 figure

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