Degeneracy of Pt-free and C⩾t-free graphs with no large complete bipartite subgraphs

A hereditary class of graphs G is χ-bounded if there exists a function f such that every graph G∈G satisfies χ(G)⩽f(ω(G)), where χ(G) and ω(G) are the chromatic number and the clique number of G, respectively. As one of the first results about χ-bounded classes, Gyárfás proved in 1985 that if G is Pt-free, i.e., does not contain a t-vertex path as an induced subgraph, then χ(G)⩽(t−1)ω(G)−1. In 2017, Chudnovsky, Scott, and Seymour proved that C⩾t-free graphs, i.e., graphs that exclude induced cycles with at least t vertices, are χ-bounded as well, and the obtained bound is again superpolynomial in the clique number. Note that Pt−1-free graphs are in particular C⩾t-free. It remains a major open problem in the area whether for C⩾t-free, or at least Pt-free graphs G, the value of χ(G) can be bounded from above by a polynomial function of ω(G). We consider a relaxation of this problem, where we compare the chromatic number with the size of a largest balanced biclique contained in the graph as a (not necessarily induced) subgraph. We show that for every t there exists a constant c such that for every ℓ and every C⩾t-free graph which does not contain Kℓ,ℓ as a subgraph, it holds that χ(G)⩽ℓc.