List-coloring graphs on surfaces with varying list-sizes

Abstract

Let $G$ be a graph embedded on a surface $S_{\epsilon }$ with Euler genus $\epsilon >0$, and let $P\subseteq V(G)$ be a set of vertices mutually at distance at least 4 apart. Suppose all vertices of $G$ have $H(\epsilon )$-lists and the vertices of $P$ are precolored, where $H(\epsilon )=\lfloor \frac{7+\sqrt{24\epsilon +1}}{2}\rfloor$ 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.

12 pages, 1 figure