Generalizing Turán'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 Erdős and Gallai.

Source : oai:arXiv.org:1803.03240

Volume: vol. 21 no. 1, ICGT 2018

Published on: July 13, 2019

Submitted on: November 6, 2018

Keywords: Mathematics - Combinatorics