Common graphs with arbitrary connectivity and chromatic number

TL;DR

This paper constructs highly connected graphs with any desired chromatic number that minimize monochromatic copies in 2-edge colourings, advancing understanding of common graphs in graph theory.

Contribution

It proves the existence of k-connected common graphs with arbitrarily large chromatic number, answering a recent open question.

Findings

Existence of k-connected common graphs with large chromatic number

Extension of previous results on common graphs

Addresses a question posed by Král, Volec, and Wei

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 Kr\'a\v{l}, Volec, and Wei who obtained common graphs with arbitrarily large chromatic number and answers a question of theirs.