Finite Three-Colourable (0,2)-Graphs Are Bipartite

Abstract

A theorem of Payan says that a cubelike graph cannot have chromatic number exactly three. A nearby question, usually discussed as Payan's finite $(0,2)$-graph question, asks whether a finite graph in which every two distinct vertices have either zero or two common neighbours can have chromatic number exactly three. The finite hypothesis is meaningful: infinite three-chromatic $(0,2)$-graphs can be constructed \cite{Payan1992}. We prove that every finite three-colourable $(0,2)$-graph is bipartite. Thus, no finite $(0,2)$-graph has chromatic number exactly three.