Reconfiguration of vertex colouring and forbidden induced subgraphs

Abstract

The reconfiguration graph of the $k$-colourings, denoted ${\mathcal{R}}_k(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two colourings are adjacent in ${\mathcal{R}}_k(G)$ if they differ in colour on exactly one vertex. In this paper, we investigate the connectivity and diameter of ${\mathcal{R}}_{k+1}(G)$ for a $k$-colourable graph $G$ restricted by forbidden induced subgraphs. We show that ${\mathcal{R}}_{k+1}(G)$ is connected for every $k$-colourable $H$-free graph $G$ if and only if $H$ is an induced subgraph of $P_4$ or $P_3+P_1$. We also start an investigation into this problem for classes of graphs defined by two forbidden induced subgraphs. We show that if $G$ is a $k$-colourable ($2K_2$, $C_4$)-free graph, then ${\mathcal{R}}_{k+1}(G)$ is connected with diameter at most $4n$. Furthermore, we show that ${\mathcal{R}}_{k+1}(G)$ is connected for every $k$-colourable ($P_5$, $C_4$)-free graph $G$.

10 pages

Cited by in corpus (2)

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