Tur\'an number of an induced complete bipartite graph plus an odd cycle

TL;DR

This paper determines the maximum number of edges in graphs that avoid certain induced bipartite subgraphs and odd cycles, providing asymptotic bounds that extend previous results and answer open questions in extremal graph theory.

Contribution

It establishes new asymptotic bounds for the Turán number of induced complete bipartite graphs plus an odd cycle, improving and generalizing prior results.

Findings

Asymptotic maximum edges for specific induced bipartite graphs and odd cycles.

Extension of previous extremal graph theory results.

Resolution of an open question by Loh, Tait, and Timmons.

Abstract

Let $k \ge 2$ be an integer. We show that if $s = 2$ and $t \ge 2$, or $s = t = 3$, then the maximum possible number of edges in a $C_{2k+1}$-free graph containing no induced copy of $K_{s,t}$ is asymptotically equal to $(t - s + 1)^{1/s}\left(\frac{n}{2}\right)^{2-1/s}$ except when $k = s = t = 2$. This strengthens a result of Allen, Keevash, Sudakov and Verstra\"{e}te and answers a question of Loh, Tait and Timmons.