Polynomial bounds for centered colorings on proper minor-closed graph classes

Abstract

For p ∈ N, a coloring λ of the vertices of a graph G is p-centered if for every connected subgraph H of G, either H receives more than p colors under λ or there is a color that appears exactly once in H. Centered colorings play an important role in the theory of sparse graphs introduced by Nešetřil and Ossona de Mendez [27], as they structurally characterize classes of bounded expansion, one of the key notions in this theory. More precisely, a class of graphs C has bounded expansion if and only if there is a function f : N → N such that every graph G ∈ C for every p ∈ N admits a p-centered coloring with at most f(p) colors. Unfortunately, known proofs of the existence of such colorings yield large upper bounds on the function f governing the number of colors needed, even for as simple classes as planar graphs. We prove that every Kt-minor-free graph admits a p-centered coloring with O(pg(t)) colors for some function g. In the special case that the graph is embeddable in a fixed surface Σ we show that it admits a p-centered coloring with O(p19) colors, with the degree of the polynomial independent of the genus of Σ. This provides the first polynomial upper bounds on the number of colors needed in p-centered colorings of graphs drawn from proper minor-closed classes, which answers an open problem posed by Dvořák [1]. As an algorithmic application, we use our main result to prove that if C is a fixed proper minor-closed class of graphs, then given graphs H and G, on p and n vertices, respectively, where G ∈ C, it can be decided whether H is a subgraph of G in time 2O(p log p) · nO(1) and space nO(1).