Five-list-coloring graphs on surfaces II. A linear bound for critical graphs in a disk

TL;DR

This paper proves a linear bound on the size of critical subgraphs in planar graphs with list coloring constraints, confirming a conjecture and advancing understanding of graph coloring on surfaces.

Contribution

It establishes a linear bound for critical subgraphs in planar graphs with list coloring, confirming a conjecture and supporting further surface-embedded graph studies.

Findings

Critical subgraph size is at most 19 times the cycle length.

Supports isoperimetric inequalities in surface-embedded graph coloring.

Facilitates future research in list coloring on surfaces.

Abstract

Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be a family of sets such that $|L(v)|\ge 5$ for every $v\in V(G)$. By an $L$-coloring of a subgraph $J$ of $G$ we mean a (proper) coloring $\phi$ of $J$ such that $\phi(v)\in L(v)$ for every vertex $v$ of $J$. We prove a conjecture of Dvorak et al. that if $H$ is a minimal subgraph of $G$ such that $C$ is a subgraph of $H$ and every $L$-coloring of $C$ that extends to an $L$-coloring of $H$ also extends to an $L$-coloring of $G$, then $|V(H)|\le 19|V(C)|$. This is a lemma that plays an important role in subsequent papers, because it motivates the study of graphs embedded in surfaces that satisfy an isoperimetric inequality suggested by this result. Such study turned out to be quite profitable for the subject of list coloring graphs on surfaces.