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 $G$.

Figures (3)

• Figure 1: The 11 graphs on 4 vertices.
• Figure 2: A $2p$-colouring of $G_p$ and a frozen $3p$-colouring.
• Figure 3: A frozen $p$-colouring of the graph $B_p$. cereceda2008

Theorems & Definitions (28)

• Theorem 1: biedl2021
• Theorem 2: feghali2021
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Lemma 1
• proof