Distant 2-Colored Components on Embeddings Part III: The General Case

TL;DR

This paper generalizes Thomassen's 5-choosability theorem for graphs on surfaces, allowing distant precolored components with specific list assignments, face-width, and separation conditions, extending previous restricted cases.

Contribution

It proves the most general case of a conjecture extending Thomassen's 5-choosability theorem for embedded graphs with distant precolored components.

Findings

Proves the general case of the conjecture for graphs on surfaces.

Extends previous results to include all embeddings with specified face-width.

Provides a comprehensive framework for list-coloring graphs with distant precolored components.

Abstract

This is the third in a sequence of three papers in which we prove the following generalization of Thomassen's 5-choosability theorem: Let $G$ be a finite graph embedded on a surface of genus $g$. Then $G$ can be $L$-colored, where $L$ is a list-assignment for $G$ in which every vertex has a 5-list except for a collection of pairwise far-apart components, each precolored with an ordinary 2-coloring, as long as the face-width of $G$ is at least $2^{\Omega(g)}$ and the precolored components are of distance at least $2^{\Omega(g)}$ apart. This provides an affirmative answer to a generalized version of a conjecture of Thomassen and also generalizes a result from 2017 of Dvo\v{r}\'ak, Lidick\'y, Mohar, and Postle about distant precolored vertices. In a previous paper, we proved that the above result holds for a restricted class of embeddings which have no separating cycles of length three or four. In this paper, we use this special case to prove that the result holds in the general case.