Odd colourings, conflict-free colourings and strong colouring numbers

The odd chromatic number and the conflict-free chromatic number are new graph parameters introduced by Petru\v{s}evski and \v{S}krekovski [2021] and Fabrici, Lu\v{z}ar, Rindo\v{s}ov\'a and Sot\'ak [2022] respectively. In this note, we show that graphs with bounded $2$-strong colouring number have bounded odd chromatic number and bounded conflict-free chromatic number. This implies that graph classes with bounded expansion have bounded odd chromatic number and bounded conflict-free chromatic number. Moreover, it follows by known results that the odd chromatic number and the conflict-free chromatic number of $k$-planar graphs is $O(k)$ which improves a recent result of Dujmovi\'{c}, Morin and Odak [2022].