Odd Colourings of Graph Products

TL;DR

This paper introduces bounds on the odd colouring number for graphs with product structure, including $k$-planar graphs, thereby answering a previously open question in graph theory.

Contribution

It establishes that graphs with certain product structures have bounded odd-colouring numbers, notably applying this to $k$-planar graphs.

Findings

Graphs with product structure have bounded odd-colouring number.

$k$-planar graphs have bounded odd-colouring number.

Answers an open question by Cranston, Lafferty, and Song.

Abstract

The odd colouring number is a new graph parameter introduced by Petru\v{s}evski and \v{S}krekovski. In this note, we show that graphs with so called product structure have bounded odd-colouring number. By known results on the product structure of $k$-planar graphs, this implies that $k$-planar graphs have bounded odd-colouring number, which answers a question of Cranston, Lafferty, and Song.