Odd colouring on the torus

Abstract

A proper vertex-colouring of a simple graph $G$ is said to be odd if, for every non-isolated vertex $v$ of $G$, some colour appears an odd number of times in the neighbourhood of $v$. We show that if $G$ embeds in the torus, then it admits a proper odd vertex-colouring with at most $9$ colours.

17 pages, 17 figures