Extensions of Thomassen's Theorem to Paths of Length At Most Four: Part II

TL;DR

This paper extends Thomassen's theorem to paths of length up to four in planar graphs, focusing on list-colorings and partial colorings of the outer cycle, with implications for coloring embeddings on surfaces.

Contribution

It proves new results on extending partial list-colorings in planar graphs with specific path and list size conditions, advancing the understanding of list-colorings in surface embeddings.

Findings

Results on extending partial $L$-colorings of the outer cycle.

Analysis of special cases with endpoint-only or partial coloring of the path.

Foundations for future work on list-colorings of high-representativity embeddings.

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. This is the second paper in a sequence of three papers in which 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 $G$, and, in particular, some useful results about the special case in which $\textrm{dom}(\phi)$ consists only of the endpoints of $P$. We also prove some results about the other special case in which $\phi$ is allowed to color some vertices of $C\setminus\mathring{P}$ but we avoid taking too many colors away from the leftover vertices of $\mathring{P}\setminus\textrm{dom}(\phi)$. We use these results in a later sequence of papers to prove some results about list-colorings of high-representativity embeddings on surfaces.