Polynomial bounds for chromatic number II: Excluding a star‐forest
Alex Scott, Paul D. Seymour, Sophie Spirkl · Journal of Graph Theory · 2022
Abstract The Gyárfás–Sumner conjecture says that for every forest , there is a function such that if is ‐free then (where are the chromatic number and the clique number of ). Louis Esperet conjectured that, whenever such a statement holds, can be chosen to be a polynomial. The Gyárfás–Sumner conjecture is only known to be true for a modest set of forests , and Esperet's conjecture is known to be true for almost no forests. For instance, it is not known when is a five‐vertex path. Here we prove Esperet's conjecture when each component of is a star.