Polynomial bounds for chromatic number II: Excluding a star-forest

Abstract

The Gyárfás–Sumner conjecture says that for every forest H, there is a function fH such that if G is H-free then x(G) ≤ fH(w(G)) (where x,w are the chromatic number and the clique number of G). Louis Esperet conjectured that, whenever such a statement holds, fH can be chosen to be a polynomial. The Gyárfás–Sumner conjecture is only known to be true for a modest set of forests H, and Esperet's conjecture is known to be true for almost no forests. For instance, it is not known when H is a five-vertex path. Here we prove Esperet's conjecture when each component of H is a star.