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

TL;DR

This paper improves bounds on improper colorings of graphs without odd clique minors, showing they can be colored with fewer colors and smaller monochromatic components than previously known.

Contribution

It proves a new bound for coloring graphs with no odd $K_t$-minor, extending previous results for $K_t$-minor free graphs and improving the size of monochromatic components.

Findings

Graphs with no odd $K_t$-minor are $(2t-2)$-colorable with small monochromatic components.

The new bounds are nearly optimal and improve previous results.

The proof combines existing methods with new ideas for odd minors.

Abstract

Strengthening Hadwiger's conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$-minor is properly $(t-1)$-colorable, 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 $(2t-2)$-coloring such that all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$. The bound on the number of colors is optimal up to a factor of $2$, improves previous bounds for the same problem by Kawarabayashi (2008), Kang and Oum (2019), Liu and Wood (2021), and strengthens a result by van den Heuvel and Wood (2018), 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 non-explicit. 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.