Five-list-coloring graphs on surfaces III. One list of size one and one list of size two

TL;DR

This paper characterizes conditions for list-coloring planar graphs with one vertex having a singleton list and another having a list of size two, extending previous work on list-colorability with precolored vertices.

Contribution

It provides a precise characterization of when such graphs are list-colorable under the specified list size constraints, advancing understanding of list-coloring with precolored vertices.

Findings

Characterization of list-colorability with one singleton and one size-two list

Extension of Thomassen's theorem to non-adjacent precolored vertices

Foundation for bounding sizes of non-colorable planar graphs with precolored cycles

Abstract

Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be a family of non-empty sets. By an $L$-coloring of $G$ we mean a (proper) coloring $\phi$ of $G$ such that $\phi(v)\in L(v)$ for every vertex $v$ of $G$. Thomassen proved that if $v_1,v_2\in V(C)$ are adjacent, $L(v_1)\ne L(v_2)$, $|L(v)|\ge3$ for every $v\in V(C)-\{v_1,v_2\}$ and $|L(v)|\ge5$ for every $v\in V(G)-V(C)$, then $G$ has an $L$-coloring. What happens when $v_1$ and $v_2$ are not adjacent? Then an $L$-coloring need not exist, but in the first paper of this series we have shown that it exists if $|L(v_1)|,|L(v_2)|\ge2$. Here we characterize when an $L$-coloring exists if $|L(v_1)|\ge1$ and $|L(v_2)|\ge2$. This result is a lemma toward a more general theorem along the same lines, which we will use to prove that minimally non-$L$-colorable planar graphs with two precolored cycles of bounded length are of bounded size. The latter result has a number of applications which we pursue elsewhere.