Connectivity of Kronecker products by K2

TL;DR

This paper determines the connectivity of the Kronecker product of a graph with K2, revealing a formula involving the original graph's connectivity and specific bipartite subgraph conditions.

Contribution

It provides a precise formula for the connectivity of G×K2 based on the original graph G's properties and bipartite subgraph structures, advancing understanding of graph product connectivity.

Findings

Connectivity of G×K2 equals min{2κ(G), a specific bipartite-based measure}

Characterization of bipartite components after vertex removal

New formula linking original graph connectivity to Kronecker product connectivity

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 $x\in X$.