Polynomial bounds for chromatic number VI. Adding a four-vertex path

Abstract

A hereditary class of graphs is χ-bounded if there is a function f such that every graph G in the class has chromatic number at most f(ω(G)), where ω(G) is the clique number of G; and the class is polynomiallyχ-bounded if f can be taken to be a polynomial. The Gyárfás–Sumner conjecture asserts that, for every forest H, the class of H-free graphs (graphs with no induced copy of H) is χ-bounded. Let us say a forest H is good if it satisfies the stronger property that the class of H-free graphs is polynomially χ-bounded.Very few forests are known to be good: for example, the goodness of the five-vertex path is open. Indeed, it is not even known that if every component of a forest H is good then H is good, and in particular, it was not known that the disjoint union of two four-vertex paths is good. Here we show the latter (with corresponding polynomial ω(G)16); and more generally, that if H is good then so is the disjoint union of H and a four-vertex path. We also prove an even more general result: if every component of H1 is good, and H2 is any path (or broom) then the class of graphs that are both H1-free and H2-free is polynomially χ-bounded.