Common graphs with arbitrary connectivity and chromatic number

Abstract

A graph H is 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. 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 Kráľ, Volec, and Wei who obtained common graphs with arbitrarily large chromatic number and answers a question of theirs.