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The odd chromatic number and the conflict-free chromatic number are new graphparameters 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, weshow that graphs with bounded $2$-strong colouring number have bounded oddchromatic number and bounded conflict-free chromatic number. This implies thatgraph classes with bounded expansion have bounded odd chromatic number andbounded conflict-free chromatic number. Moreover, it follows by known resultsthat 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].

Odd colourings, conflict-free colourings and strong colouring numbers