List-coloring graphs on surfaces with varying list-sizes
Alice M. Dean, Joan P. Hutchinson

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 be a graph embedded on a surface with Euler genus , and let be a set of vertices mutually at distance at least 4 apart. Suppose all vertices of have -lists and the vertices of are precolored, where is the Heawood number. We show that the coloring of extends to a list-coloring of 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.
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory
