Optimal List Recoloring of Subcubic Graphs and Complete Multipartite Graphs

TL;DR

This work addresses the diameter of the $L$-recoloring graph $C_L(G)$ under the standard list condition $|L(v)| \ge d(v)+2$, focusing on two graph families. Using a minimal-counterexample framework and color-shift digraph techniques, the authors derive recoloring sequences that reduce to smaller instances. They prove the conjectured bound $\mathrm{diam}(C_L(G)) \le n(G) + \mu(G)$ for both subcubic graphs and complete multipartite graphs, matching known lower bounds in these classes. The results advance the understanding of reconfiguration diameters for list-colorings and extend the landscape of graphs for which Cambie et al.'s conjecture holds, with potential implications for related reconfiguration problems.

Abstract

For a list-assignment $L$, the reconfiguration graph $C_L(G)$ of a graph $G$ is the graph whose vertices are proper $L$-colorings of $G$ and whose edges link two colorings that differ on only one vertex. If $|L(v)| \ge d(v) + 2$ for every vertex of $G$, it is known that $C_L(G)$ is connected. In this case, Cambie et al. investigated the diameter of $C_L(G)$. They conjectured that $diam(C_L(G)) \le n(G) + μ(G)$ with $μ(G)$ the size of a maximum matching of $G$ and proved several results towards this conjecture. We answer to two of their open problems by proving the conjecture for two classes of graphs, namely subcubic graphs and complete multipartite graphs.

Figures (7)

• Figure 1: Two colorings $\alpha$ (at the centers) and $\beta$ (on the borders) of a graph $G$, the color-shift digraph $\overrightarrow{D_{\alpha,\beta}}$ and multigraph $D_{\alpha, \beta}$ of $G$.
• Figure 2: Recoloring sequence of Lemma \ref{['lem:cc2']} for three consecutive vertices $u,v$ and $w$ in $\overrightarrow{D_{\alpha, \beta}}$. We only represent some arcs of the color-shift digraph.
• Figure 3: Reconfiguration sequence of Lemma \ref{['cl:agreeV0']}. First, we make the colorings agree on $W$ (in gray here) on colors distinct from $c$ (violet here), then we recolor the vertices of $V_0 \setminus W$ with $c$. We only represent arcs of the color-shift digraph that are between vertices of $V_0$.
• Figure 4: Reconfiguration sequence of Lemma \ref{['prop:matchingab']} to obtain $\gamma$. The color of $v$ in $\alpha$ appears only once in $\alpha$ but at least twice in $\beta$. We only represent some arcs of the color-shift digraph.
• Figure 5: Reconfiguration sequence of Lemma \ref{['prop:no2into1']} to obtain $\gamma$. The color violet appears at least twice in $\beta$, and exactly twice in $\alpha$: on $v_1$ and $v_2$. We only represent some arcs of the color-shift digraph.

Theorems & Definitions (35)

• Conjecture 1.1: Cambie et al. cambie2024optimally
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Corollary 1.5
• Theorem 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof