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

TL;DR

This paper investigates bounds on the chromatic number of certain classes of graphs that exclude large paths, cycles, and complete bipartite subgraphs, providing new polynomial bounds under specific conditions.

Contribution

It establishes polynomial bounds on the chromatic number for $C_{ geq t}$-free graphs excluding large bicliques, advancing understanding of $ ext{chi}$-boundedness in these classes.

Findings

For every $t$, there exists a constant $c$ such that $C_{ geq t}$-free graphs without $K_{ ext{ell,ell}}$ subgraphs satisfy $ ext{chi}(G) \

The bounds are polynomial in the size of the largest biclique, providing new insights into chromatic bounds for restricted graph classes.

The results extend previous work on $ ext{chi}$-boundedness by considering non-induced subgraph exclusions and biclique constraints.

Abstract

A hereditary class of graphs $\mathcal{G}$ is \emph{$\chi$-bounded} if there exists a function $f$ such that every graph $G \in \mathcal{G}$ satisfies $\chi(G) \leq f(\omega(G))$, where $\chi(G)$ and $\omega(G)$ are the chromatic number and the clique number of $G$, respectively. As one of the first results about $\chi$-bounded classes, Gy\'{a}rf\'{a}s proved in 1985 that if $G$ is $P_t$-free, i.e., does not contain a $t$-vertex path as an induced subgraph, then $\chi(G) \leq (t-1)^{\omega(G)-1}$. In 2017, Chudnovsky, Scott, and Seymour proved that $C_{\geq t}$-free graphs, i.e., graphs that exclude induced cycles with at least $t$ vertices, are $\chi$-bounded as well, and the obtained bound is again superpolynomial in the clique number. Note that $P_{t-1}$-free graphs are in particular $C_{\geq t}$-free. It remains a major open problem in the area whether for $C_{\geq t}$-free, or at least $P_t$-free graphs $G$, the value of $\chi(G)$ can be bounded from above by a polynomial function of $\omega(G)$. 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 $C_{\geq t}$-free graph which does not contain $K_{\ell,\ell}$ as a subgraph, it holds that $\chi(G) \leq \ell^{c}$.