List-coloring graphs on surfaces with varying list-sizes

TL;DR

This paper proves that precolorings of well-separated vertices in surface-embedded graphs with lists of size determined by the Heawood number can be extended to full list-colorings, generalizing planar results to surfaces.

Contribution

It establishes the extendability of precolorings in surface-embedded graphs with varying list sizes, generalizing planar graph coloring results to surfaces.

Findings

Precoloring extension is possible with distance at least 4.

The distance bound of 4 is optimal.

Results generalize planar graph coloring to surfaces.

Abstract

Let $G$ be a graph embedded on a surface $S_\varepsilon$ with Euler genus $\varepsilon > 0$, and let $P\subseteq V(G)$ be a set of vertices mutually at distance at least 4 apart. Suppose all vertices of $G$ have $H(\varepsilon)$-lists and the vertices of $P$ are precolored, where $H(\varepsilon)=\Big\lfloor\frac{7 + \sqrt{24\varepsilon + 1}}{2}\Big\rfloor$ is the Heawood number. We show that the coloring of $P$ extends to a list-coloring of $G$ and that the distance bound of 4 is best possible. Our result provides an answer to an analogous question of Albertson about extending a precoloring of a set of mutually distant vertices in a planar graph to a 5-list-coloring of the graph and generalizes a result of Albertson and Hutchinson to list-coloring extensions on surfaces.