Distant 2-Colored Components on Embeddings Part I: Connecting Faces

TL;DR

This paper generalizes Thomassen's 5-choosability theorem for graphs on surfaces, allowing for distant precolored components with 2-colorings, under certain face-width and distance conditions.

Contribution

It proves a new theorem extending 5-choosability to graphs with distant precolored components on surfaces, answering a conjecture and building on prior results.

Findings

Graphs with distant precolored components are 5-choosable under face-width and distance conditions.

The result generalizes previous theorems about precolored vertices to components.

Provides a foundation for further extensions in graph coloring on surfaces.

Abstract

This is the first 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 $2^{\Omega(g)}$ and the precolored components are of distance $2^{\Omega(g)}$ 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.