Improved bound for improper colourings of graphs with no odd clique minor

Abstract

Strengthening Hadwiger’s conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$ -minor is properly $(t-1)$ -colourable. This is known as the Odd Hadwiger’s conjecture. We prove a relaxation of the above conjecture, namely we show that every graph with no odd $K_t$ -minor admits a vertex $(2t-2)$ -colouring such that all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$ . The bound on the number of colours is optimal up to a factor of $2$ , improves previous bounds for the same problem by Kawarabayashi (2008, Combin. Probab. Comput.17 815–821), Kang and Oum (2019, Combin. Probab. Comput.28 740–754), Liu and Wood (2021, arXiv preprint, arXiv:1905.09495), and strengthens a result by van den Heuvel and Wood (2018, J. Lond. Math. Soc.98 129–148), who showed that the above conclusion holds under the more restrictive assumption that the graph is $K_t$ -minor-free. In addition, the bound on the component-size in our result is much smaller than those of previous results, in which the dependency on $t$ was given by a function arising from the graph minor structure theorem of Robertson and Seymour. Our short proof combines the method by van den Heuvel and Wood for $K_t$ -minor-free graphs with some additional ideas, which make the extension to odd $K_t$ -minor-free graphs possible.