Degeneracy of $P_t$-free and $C_{\geq t}$-free graphs with no large complete bipartite subgraphs

Abstract

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 Ptfree, 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 and every C t -free graph which does not contain K , as a subgraph, it holds that χ(G)c .

Polynomial bounds for chromatic number. III. Excluding a double star

“…Since |L i | ≥ ω s+5 , v has a neighbour in L i ; and so from the choice of the pair i, j, v has at most ω s + ω s+3 non-neighbours in L j . This proves (1).…”

Polynomial bounds for chromatic number. I. Excluding a biclique and an induced tree

“…Then G[Z] is H ′ -free (because otherwise there would be a vertex v ∈ Z, and a subset X ⊆ Z \ {v}, and an isomorphism from H ′ to G[X ∪ {v}] mapping q to v, and hence with X ∩ Y v = ∅; but no vertex of Y v belongs to Z, since Z is stable in J ′ ). From the inductive hypothesis, χ(Z) ≤ c ′ t, and hence (1). This proves 4.2.…”

“…Let ψ(r) = v, and for 1 ≤ i ≤ t η and each v ∈ V (H i ), define ψ(v) = φ ′ i (w) where w is the parent of v in (H, r). Then ψ is a (t, η)-infusion of (H, r) into G, with root v. This proves (1).…”

“…Finally, there is a second pretty theorem in the paper [1] of Bonamy, Pilipczuk, Rzazewski, Thomassé and Walczak:…”

“…What about the analogue of Esperet's question: do 1.2 and 1.3 remain true if we require f to be a polynomial in τ (G)? This question was raised by Bonamy, Bousquet, Pilipczuk, Rzazewski, Thomassé and Walczak in [1], and they proved it when H is a path, that is:…”