An alternate proof of Payan's theorem on cubelike graphs

A cubelike graph is a Cayley graph on the product ${\mathbb{Z}}_2\times \cdots \times {\mathbb{Z}}_2$ of the integers modulo $2$ with itself finitely many times. In 1992, Payan proved that no cubelike graph can have chromatic number $3$. The authors of the present paper previously developed a general matrix method for studying chromatic numbers of Cayley graphs on abelian groups. In this note, we apply this method of Heuberger matrices to give an alternate proof of Payan’s theorem.