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 known as the Odd conjecture . We prove a relaxation of above namely we show admits vertex $(2t-2)$ -colouring such all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$ The bound on number colours optimal up to factor $2$ , improves previous bounds for same problem by Kawarabayashi (2008, Combin. Probab. Comput. 17 815–821), Kang Oum (2019, 28 740–754), Liu Wood (2021, arXiv preprint, arXiv:1905.09495), strengthens result van den Heuvel (2018, J. Lond. Math. Soc. 98 129–148), who showed conclusion holds under more restrictive assumption -minor-free. In addition, component-size our much smaller than those results, which dependency $t$ was given function arising from minor structure theorem Robertson Seymour. Our short proof combines method -minor-free graphs some additional ideas, make extension possible.