Odd Colourings of Graph Products

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.