The odd chromatic number of a planar graph is at most 8

TL;DR

This paper proves that every planar graph can be properly coloured with at most 8 colours such that each non-isolated vertex has an odd number of neighbours of some colour, improving previous bounds.

Contribution

It establishes that the odd chromatic number of any planar graph is at most 8, resolving a conjecture and extending prior results.

Findings

All planar graphs have an odd chromatic number of at most 8.

Previous bounds of 9 colours are improved to 8.

The result applies universally to all planar graphs.

Abstract

Petru\v{s}evski and \v{S}krekovski \cite{odd9} recently introduced the notion of an odd colouring of a graph: a proper vertex colouring of a graph $G$ is said to be \emph{odd} if for each non-isolated vertex $x \in V(G)$ there exists a colour $c$ appearing an odd number of times in $N(x)$. Petru\v{s}evski and \v{S}krekovski proved that for any planar graph $G$ there is an odd colouring using at most $9$ colours and, together with Caro \cite{oddremarks}, showed that $8$ colours are enough for a significant family of planar graphs. We show that $8$ colours suffice for all planar graphs.