Weak diameter coloring of graphs on surfaces

Abstract

Consider a graph G drawn on a fixed surface, and assign to each vertex a list of colors of size at least two if G is triangle-free and at least three otherwise. We prove that we can give each vertex a color from its list so that each monochromatic connected subgraph has bounded weak diameter (i.e., diameter measured in the metric of the whole graph G, not just the subgraph). In case that G has bounded maximum degree, this implies that each connected monochromatic subgraph has bounded size. This solves a problem of Esperet and Joret for planar triangle-free graphs, and extends known results in the general case to the list setting, answering a question of Wood.