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arXiv · 1803.03240

The maximum number of $P_\ell$ copies in $P_k$-free graphs

Ervin Győri·Nika Salia·Casey Tompkins·Oscar Zamora
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Abstract

Generalizing Tur\'an's classical extremal problem, Alon and Shikhelman investigated the problem of maximizing the number of $T$ copies in an $H$-free graph, for a pair of graphs $T$ and $H$. Whereas Alon and Shikhelman were primarily interested in determining the order of magnitude for large classes of graphs $H$, we focus on the case when $T$ and $H$ are paths, where we find asymptotic and in some cases exact results. We also consider other structures like stars and the set of cycles of length at least $k$, where we derive asymptotically sharp estimates. Our results generalize well-known extremal theorems of Erd\H{o}s and Gallai.

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Ervin Győri, Nika Salia, Casey Tompkins, Oscar Zamora. 2018-03-08. The maximum number of $P_\ell$ copies in $P_k$-free graphs. https://doi.org/10.23638/dmtcs-21-1-14

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