List-coloring embedded graphs
Zdenĕk Dvořák, Ken‐ichi Kawarabayashi · 2013
For any fixed surface σ of genus g, we give an algorithm to decide whether a graph G of girth at least five embedded in σ 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.