Induced subdivisions in $K_{s,s}$-free graphs with polynomial average degree

TL;DR

This paper establishes tight bounds on the conditions under which graphs contain induced subdivisions of a given graph or specific bipartite subgraphs, advancing understanding of graph structure related to forbidden subgraphs.

Contribution

It proves a conjecture on induced subdivisions in $K_{s,s}$-free graphs with polynomial average degree, and generalizes results to broader classes of graphs with tight bounds.

Findings

Graphs with high average degree contain induced subdivisions of any fixed graph.

Graphs with high average degree contain large bipartite subgraphs without 4-cycles.

Results are tight and confirm conjectures in graph theory.

Abstract

In this paper we prove that for every $s\geq 2$ and every graph $H$ the following holds. Let $G$ be a graph with average degree $\Omega_H(s^{C|H|^2})$, for some absolute constant $C>0$, then $G$ either contains a $K_{s,s}$ or an induced subdivision of $H$. This is essentially tight and confirms a conjecture of Bonamy, Bousquet, Pilipczuk, Rz\k{a}\.zewski, Thomass\'e, and Walczak. A slightly weaker form of this has been independently proved by Bourneuf, Buci\'c, Cook, and Davies. We actually prove a much more general result which implies the above (with worse dependence on $|H|$). We show that for every $ k\geq 2$ there is $C_k>0$ such that any graph $G$ with average degree $s^{C_k}$ either contains a $K_{s,s}$ or an induced subgraph $G'\subseteq G$ without $C_4$'s and with average degree at least $k$. Finally, using similar methods we can prove the following. For every $k,t\geq 2$ every graph $G$ with average degree at least $C_tk^{\Omega(t)}$ must contain either a $K_k$, an induced $K_{t,t}$ or an induced subdivision of $K_k$. This is again essentially tight up to the implied constants and answers in a strong form a question of Davies.