List-Coloring Graphs on Surfaces with Varying List-Sizes

Let G be a graph embedded on a surface Sε with Euler genus ε > 0, and let P ⊂ V (G) be a set of vertices mutually at distance at least 4 apart. Suppose all vertices of G have H(ε)-lists and the vertices of P are precolored, where H(ε) = ⌊ 7+ √ 24ε+1 2 ⌋ 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.