Disjoint Correspondence Colorings for $K_5$-Minor-free Graphs

TL;DR

This paper investigates disjoint correspondence colorings in $K_5$-minor-free graphs, proving the existence of three disjoint colorings under certain conditions and providing counterexamples for others.

Contribution

It establishes that every $K_5$-minor-free graph admits three disjoint $ extbf{M}$-colorings for any correspondence 6-cover, advancing understanding of coloring variants in minor-free graphs.

Findings

Existence of three disjoint $ extbf{M}$-colorings for 6-covers

Counterexamples for three disjoint colorings with 5-covers

Extension of coloring results beyond planar graphs

Abstract

Thomassen famously proved that every planar graph is 5-choosable. We explore variants of this result, focusing on finding disjoint correspondence colorings, in the more general class of $K_5$-minor-free graphs. Correspondence colorings generalize list colorings as follows. Given a graph $G$ and a positive integer $t$, a correspondence $t$-cover $\textbf{M}$ assigns to each $v\in V(G)$ a set of allowable colors $\{1_v,\ldots,t_v\}$ and to each edge $vw\in E(G)$ a matching between $\{1_v,\ldots,t_v\}$ and $\{1_w,\ldots,t_w\}$. An $\textbf{M}$-coloring $\varphi$ picks for each vertex $v$ a color $\varphi(v)$ (from the set $\{1_v,\ldots,t_v\}$) such that for each edge $vw\in E(G)$ the colors $\varphi(v),\varphi(w)$ are not matched to each other. Two $\textbf{M}$-colorings $\varphi_1,\varphi_2$ of $G$ are called disjoint if $\varphi_1(v)\ne\varphi_2(v)$ for all $v\in V(G)$. For every $K_5$-minor-free graph $G$ and every correspondence 6-cover $\textbf{M}$ of $G$, we construct 3 pairwise disjoint $\textbf{M}$-colorings $\varphi_1,\varphi_2,\varphi_3$. In contrast, we provide examples of $K_5$-minor-free graphs and correspondence 5-covers $\textbf{M}$ that do not admit 3 disjoint $\textbf{M}$-colorings.