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arXiv · 2311.18301

Rainbow common graphs must be forests

Yihang Sun
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Abstract

We study the rainbow version of the graph commonness property: a graph $H$ is $r$-rainbow common if the number of rainbow copies of $H$ (where all edges have distinct colors) in an $r$-coloring of edges of $K_n$ is maximized asymptotically by independently coloring each edge uniformly at random. $H$ is \emph{$r$-rainbow uncommon} otherwise. We show that if $H$ has a cycle, then it is $r$-rainbow uncommon for every $r$ at least the number of edges of $H$. This generalizes a result of Erdős and Hajnal, and proves a conjecture of De Silva, Si, Tait, Tunçbilek, Yang, and Young.

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Yihang Sun. 2024-07-10. Rainbow common graphs must be forests. https://arxiv.org/abs/2311.18301

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