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

TL;DR

The paper investigates disjoint correspondence colorings on $K_5$-minor-free graphs, proving that every graph in this class admits three pairwise disjoint $\textbf{M}$-colorings for any correspondence 6-cover $\textbf{M}$, while a 5-cover can fail this property. The authors combine Wagner’s decomposition, base lemmas for $M_8$ and planar graphs, and a Thomassen-style inductive approach (a Key Lemma employing Hall’s theorem) to extend colorings through 2-sums and 3-sums. A counterexample on a $K_5$-minor-free graph with a 5-cover illustrates tightness of the 6-cover condition. The results contribute to the broader study of correspondence packing and weighted flexiblity in planar and near-planar graphs, and suggest directions for tightening bounds on $\chi^{\star}_c(\mathcal{K}_5)$. Overall, the work advances understanding of disjoint colorings in minor-free graphs and informs conjectures about flexibility in colorings.

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.

Figures (1)

• Figure 1: Left: The portion of $\textbf{M}$ induced by $N[w_n]$. Right: The portion induced by $N[w_n]-\{w_n\}$. (Each thick gray line, say between vertices $x$ and $y$, denotes a perfect matching between $[6]_x$ and $[6]_y$. All other aspects of the figure are explained in \ref{['example1']}.)

Theorems & Definitions (13)

• Conjecture 1
• Lemma A: CCvBDK
• proof
• Theorem B: wagner
• Lemma 2
• proof
• Lemma 3
• proof : Proof of the Main Theorem.
• proof : Proof of \ref{['planar-lem']}.
• proof : Proof of the Key Lemma.