Some Extensions of Thomassen's Theorem to Longer Paths

TL;DR

This paper extends Thomassen's theorem by exploring list-colorings of planar graphs with specific conditions on paths and list sizes, aiming to support future research on surface embeddings.

Contribution

It introduces new results on partial list-colorings in planar graphs with paths of length up to four, advancing understanding of list-coloring extensions.

Findings

Results on partial L-colorings of cycles with paths of length up to four

Conditions under which partial colorings extend to entire graphs

Framework for future work on list-colorings on surfaces

Abstract

Let $G$ be a planar embedding with list-assignment $L$ and outer cycle $C$, and let $P$ be a path of length at most four on $C$, where each vertex of $G\setminus C$ has a list of size at least five and each vertex of $C\setminus P$ has a list of size at least three. In this paper, we prove some results about partial $L$-colorings $\phi$ of $C$ with the property that any extension of $\phi$ to an $L$-coloring of $\textrm{dom}(\phi)\cup V(P)$ extends to $L$-color all of $G$. We use these results in a later sequence of papers to prove some results about list-colorings of high-representativity embeddings on surfaces.