List-coloring embedded graphs

TL;DR

This paper presents an efficient algorithm for list-coloring graphs embedded on surfaces of fixed genus, even with precolored subgraphs, extending the applicability of coloring algorithms in topological graph theory.

Contribution

It introduces a new algorithm for 3-list-coloring embedded graphs with girth at least five, accommodating precolored subgraphs, with complexity depending on surface genus and precolored components.

Findings

Decides 3-list-colorability in linear time for fixed genus surfaces.

Handles precolored subgraphs with increased but controlled complexity.

Applicable to other coloring problems on surfaces.

Abstract

For any fixed surface Sigma of genus g, we give an algorithm to decide whether a graph G of girth at least five embedded in Sigma is colorable from an assignment of lists of size three in time O(|V(G)|). Furthermore, we can allow a subgraph (of any size) with at most s components to be precolored, at the expense of increasing the time complexity of the algorithm to O(|V(G)|^{K(g+s)+1}) for some absolute constant K; in both cases, the multiplicative constant hidden in the O-notation depends on g and s. This also enables us to find such a coloring when it exists. The idea of the algorithm can be applied to other similar problems, e.g., 5-list-coloring of graphs on surfaces.