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arXiv · 2309.12866

On triangle-free graphs maximizing embeddings of bipartite graphs

Dmitriy Gorovoy·Andrzej Grzesik·Justyna Jaworska
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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 $\Omega(x)$.

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Dmitriy Gorovoy, Andrzej Grzesik, Justyna Jaworska. 2023-09-22. On triangle-free graphs maximizing embeddings of bipartite graphs. https://arxiv.org/abs/2309.12866

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