Connectivity of Kronecker products by K2
Abstract
Let $\kappa(G)$ 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 $\kappa(G\times K_2)=\textup{min}\{2\kappa(G), \textup{min}\{|X|+2|Y|\}\}$, 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 $x\in X$.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2011
- DOI:
- 10.48550/arXiv.1106.1255
- arXiv:
- arXiv:1106.1255
- Bibcode:
- 2011arXiv1106.1255W
- Keywords:
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- Mathematics - Combinatorics;
- 05C40 05C40
- E-Print:
- 6 pages