Five-List-Coloring Graphs on Surfaces: The Many Faces Far-Apart Generalization of Thomassen's Theorem

TL;DR

This paper generalizes Thomassen's 5-list-coloring theorem for planar graphs to graphs on surfaces, allowing multiple distant faces with specific list coloring conditions, expanding the scope of colorability results.

Contribution

It extends Thomassen's theorem to graphs with multiple faces on surfaces, where faces are sufficiently far apart, broadening the applicability of list-coloring results.

Findings

Generalization of Thomassen's theorem to multiple faces

Colorability holds when faces are pairwise distant by a universal constant

Applicable to graphs embedded on surfaces beyond the plane

Abstract

Let $G$ be a plane graph with $C$ the boundary of the outer face and let $(L(v):v\in V(G))$ be a family of non-empty sets. By an $L$-coloring of a subgraph $J$ of $G$ we mean a (proper) coloring $\phi$ of $J$ such that $\phi(v)\in L(v)$ for every vertex $v$ of $J$. 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 $v\in V(C)\setminus \{v_1,v_2\}$ and $|L(v)|\ge5$ for every $v\in V(G)\setminus V(C)$, then $G$ has an $L$-coloring. As one final application in this last part of our series on $5$-list-coloring, we derive from all of our theory a far-reaching generalization of Thomassen's theorem, namely the generalization of Thomassen's theorem to arbitrarily many such faces provided that the faces are pairwise distance $D$ apart for some universal constant $D>0$.