Non-three-colorable common graphs exist

TL;DR

This paper demonstrates the existence of common graphs that are not three-colorable, providing the first such example and challenging previous conjectures about the universality of common graphs.

Contribution

It proves that the 5-wheel graph is common, offering the first example of a non-three-colorable common graph, thus disproving earlier conjectures.

Findings

The 5-wheel is a common graph.

Common graphs are more diverse than previously thought.

Not all common graphs are three-colorable.

Abstract

A graph H is called common if the total number of copies of H in every graph and its complement asymptotically minimizes for random graphs. A former conjecture of Burr and Rosta, extending a conjecture of Erdos asserted that every graph is common. Thomason disproved both conjectures by showing that the complete graph of order four is not common. It is now known that in fact the common graphs are very rare. Answering a question of Sidorenko and of Jagger, Stovicek and Thomason from 1996 we show that the 5-wheel is common. This provides the first example of a common graph that is not three-colorable.