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arXiv · 1106.1255

Connectivity of Kronecker products by K2

Wei Wang·Zhidan Yan
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Abstract

Let $κ(G)$ be the connectivity of $G$. The Kronecker product $G_1\times G_2$ of graphs $G_1$ and $G_2$ has vertex set $V(G_1\times G_2)=V(G_1)\times V(G_2)$ and edge set $E(G_1\times G_2)=\{(u_1,v_1)(u_2,v_2):u_1u_2\in E(G_1),v_1v_2\in E(G_2)\}$. In this paper, we prove that $κ(G\times K_2)=\textup{min}\{2κ(G), \textup{min}\{|X|+2|Y|\}\}$, where the second minimum is taken over all disjoint sets $X,Y\subseteq V(G)$ satisfying (1)$G-(X\cup Y)$ has a bipartite component $C$, and (2) $G[V(C)\cup \{x\}]$ is also bipartite for each $x\in X$.

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Wei Wang, Zhidan Yan. 2011-06-07. Connectivity of Kronecker products by K2. https://arxiv.org/abs/1106.1255

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