Frozen colourings in $2K_2$-free graphs

TL;DR

This work investigates frozen colourings in $2K_2$-free and related $C_4$-free graphs through the lens of reconfiguration graphs $\\mathcal{R}_k(G)$. It provides four infinite families of $2K_2$-free graphs that admit frozen colourings, and introduces an edge-subdivision operation that preserves $2K_2$-freeness while increasing the available colours and maintaining frozen colourings. Consequently, for every $k \ge 4$ there exist $k$-chromatic $2K_2$-free graphs with a frozen $(k+1)$-colouring, with further structure explored via frozen clique partitions in $C_4$-free graphs. A key positive result is that all 3-chromatic $2K_2$-free graphs are recolourable, and the paper concludes with open questions on the remaining cases and broader dichotomies for recolourability in forbidden-subgraph classes.

Abstract

The \emph{reconfiguration graph of the $k$-colourings} of a graph $G$, denoted $\mathcal{R}_k(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two vertices of $\mathcal{R}_k(G)$ are joined by an edge if the colourings of $G$ they correspond to differ in colour on exactly one vertex. A $k$-colouring of a graph $G$ is called \emph{frozen} if it is an isolated vertex in $\mathcal{R}_k(G)$; in other words, for every vertex $v \in V(G)$, $v$ is adjacent to a vertex of every colour different from its colour. A clique partition is a partition of the vertices of a graph into cliques. A clique partition is called a $k$-clique-partition if it contains at most $k$ cliques. Clearly, a $k$-colouring of a graph $G$ corresponds precisely to a $k$-clique-partition of its complement, $\overline{G}$. A $k$-clique-partition $\mathcal{Q}$ of a graph $H$ is called \emph{frozen} if for every vertex $v \in V(H)$, $v$ has a non-neighbour in each of the cliques of $\mathcal{Q}$ other than the one containing $v$. The cycle on four vertices, $C_4$, is sometimes called the \emph{square}; its complement is called $2K_2$. We give several infinite classes of $2K_2$-free graphs with frozen colourings. We give an operation which transforms a $k$-chromatic graph with a frozen $(k+1)$-colouring into a $(k+1)$-chromatic graph with a frozen $(k+2)$-colouring. Our operation preserves being $2K_2$-free. It follows that for all $k \ge 4$, there is a $k$-chromatic $2K_2$-free graph with a frozen $(k+1)$-colouring. We prove these results by studying frozen clique partitions in $C_4$-free graphs. We say a graph $G$ is \emph{recolourable} if $R_{\ell}(G)$ is connected for all $\ell$ greater than the chromatic number of $G$. We prove that every 3-chromatic $2K_2$-free graph is recolourable.

Figures (10)

• Figure 1: A square-free graph $\overline{ME_2}$ with a 4-clique-partition (left) and a frozen 5-clique-partition (right). The numbers indicate which clique a vertex is in. Equivalently, the numbers indicate a 4-colouring of the complement $ME_2$ of the graph shown (left) and a frozen 5-colouring of $ME_2$ (right).
• Figure 2: A square-free graph $\overline{ME_3}$ with a 6-clique-partition (left) and a frozen 7-clique-partition (right). The numbers indicate which clique a vertex is in. Equivalently, the numbers indicate a 6-colouring of the complement $ME_3$ of the graph shown (left) and a frozen 7-colouring of $ME_3$ (right).
• Figure 3: A square-free graph $\overline{KM_2}$ with a 4-clique-partition (left) and a frozen 5-clique-partition (right). Equivalently, a 4-colouring of the complement $KM_2$ of the graph shown (left) and a frozen 5-colouring of $KM_2$ (right).
• Figure 4: A square-free graph $\overline{KM_3}$ with a 6-clique-partition (left) and a frozen 7-clique-partition (right). Equivalently, a 6-colouring of the complement $KM_3$ of the graph shown (left) and a frozen 7-colouring of $KM_3$ (right).
• Figure 5: A $C_4$-free graph $\overline{KE_2}$ with a 4-clique-partition (left) and a frozen 6-clique-partition (right). Equivalently, a 4-colouring of the complement $KE_2$ (left) and a frozen 6-colouring (right).

Theorems & Definitions (59)

• Proposition 1
• Lemma 1: Renaming Lemma bonamy2018
• Lemma 2
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Claim 1
• Claim 2