On triangle-free graphs maximizing embeddings of bipartite graphs

TL;DR

This paper extends the understanding of extremal triangle-free graphs by identifying conditions under which bipartite graphs with certain matchings are maximized in complete bipartite graphs, refining previous results.

Contribution

It improves prior bounds by showing that bipartite graphs with specific matching and unmatched vertex constraints are maximized in complete bipartite graphs.

Findings

Maximum copies of certain bipartite graphs achieved in complete bipartite graphs

Improved bounds on unmatched vertices for extremal configurations

Counterexamples when unmatched vertices are proportional to matching size

Abstract

In 1991 Gy\H ori, Pach, and Simonovits proved that for any bipartite graph $H$ containing a matching avoiding at most 1 vertex, the maximum number of copies of $H$ in any large enough triangle-free graph is achieved in a balanced complete bipartite graph. In this paper we improve their result by showing that if $H$ is a bipartite graph containing a matching of size $x$ and at most $\frac{1}{2}\sqrt{x-1}$ unmatched vertices, then the maximum number of copies of $H$ in any large enough triangle-free graph is achieved in a complete bipartite graph. We also prove that such a statement cannot hold if the number of unmatched vertices is $\Omega(x)$.