On some problems concerning the commonality of graphs Sejin Ko

Abstract

A graph $H$ is \emph{common} if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring. Classifying common graphs is a wide-open problem. However, we make progress in different directions. We prove that, given $k,r>0$, there exists a $k$-connected common graph with chromatic number at least $r$. The result is built upon the recent breakthrough of Kr\'a\v{l}, Volec, and Wei who obtained common graphs with arbitrarily large chromatic number and answers a question of theirs. Additionally, we prove that there exists a graph with even girth which is not density common. Based on the local variant of Sidorenko's conjecture, we prove that if $H_1$ has an even girth, there exist $H_2$ and $p$ such that $(H_1,H_2)$ is $(p,1-p)$-common.