Odd colouring on the torus

TL;DR

This paper proves that any graph embeddable on a torus can be properly coloured with at most 9 colours such that each non-isolated vertex has a neighbourhood with some colour appearing an odd number of times.

Contribution

It establishes an upper bound of 9 colours for odd colourings of torus-embeddable graphs, extending understanding of graph colourings on surfaces.

Findings

Graphs on the torus admit a proper odd colouring with at most 9 colours.

The result provides a new bound in the study of odd colourings on surface-embedded graphs.

This advances the theory of graph colourings in topologically constrained settings.

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.