5-Coloring Reconfiguration of Planar Graphs with No Short Odd Cycles

TL;DR

This paper proves that for planar graphs without 3- and 5-cycles, the reconfiguration graph of proper 5-colorings has a quadratic diameter, advancing understanding of coloring reconfiguration in specific graph classes.

Contribution

It establishes an $O(n^2)$ bound on the diameter of the coloring reconfiguration graph for a new class of planar graphs with no short odd cycles.

Findings

Diameter of $oldsymbol{ ext{C}_5(G)}$ is $oldsymbol{O(n^2)}$ for planar graphs with no 3- or 5-cycles.

Supports partial validation of Cereceda's conjecture for specific planar graph classes.

Extends previous results to broader classes of planar graphs with cycle restrictions.

Abstract

The coloring reconfiguration graph $\mathcal{C}_k(G)$ has as its vertex set all the proper $k$-colorings of $G$, and two vertices in $\mathcal{C}_k(G)$ are adjacent if their corresponding $k$-colorings differ on a single vertex. Cereceda conjectured that if an $n$-vertex graph $G$ is $d$-degenerate and $k\geq d+2$, then the diameter of $\mathcal{C}_k(G)$ is $O(n^2)$. Bousquet and Heinrich proved that if $G$ is planar and bipartite, then the diameter of $\mathcal{C}_5(G)$ is $O(n^2)$. (This proves Cereceda's Conjecture for every such graph with degeneracy 3.) They also highlighted the particular case of Cereceda's Conjecture when $G$ is planar and has no 3-cycles. As a partial solution to this problem, we show that the diameter of $\mathcal{C}_5(G)$ is $O(n^2)$ for every planar graph $G$ with no 3-cycles and no 5-cycles.