A Note on the Complexity of Graph Recoloring

TL;DR

The paper addresses the complexity of deciding whether a graph is $k$-mixing, i.e., whether all $k$-colorings can be transformed into one another via single-vertex recolorings while maintaining a proper coloring. Building on the known result that $3$-Mixing is co-NP-complete, it proves that for every $k \ge 4$, $k$-Mixing is co-NP-hard, even on graphs that are $(k-1)$-colorable. The proof uses a reduction from the $3$-To-$2$ problem on bipartite graphs to $k$-Mixing by constructing a graph $G$ from a bipartite base $B$ augmented with a $(k-3)$-clique fully connected to $B$, establishing that $G$ is $k$-mixing iff every $3$-coloring of $B$ can be transformed into a $2$-coloring. A key lemma connects $3$-mixing to reachability of a $2$-coloring in bipartite graphs, enabling the reduction; the paper also discusses a conjecture that $4$-To-$3$ is PSPACE-complete and notes the potential PSPACE-hardness implications for related problems.

Abstract

We say that a graph is $k$-mixing if it is possible to transform any $k$-coloring into any other via a sequence of single vertex recolorings keeping a proper coloring all along. Cereceda, van den Heuvel and Johnson proved that deciding if a graph is $3$-mixing is co-NP-complete and left open the case $k \ge 4$. We prove that for every $k \ge 4$, $k$-mixing is co-NP-hard.

Theorems & Definitions (7)

• Theorem 1: Cereceda, van den Heuvel, Johnson CerecedaHJ09
• Theorem 2
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['thm:4+mixing']}
• Conjecture 1: Cereceda, van den Heuvel, Johnson CerecedaHJ09
• Conjecture 2