Rainbow Common Graphs Must Be Forests

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 $r$-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.