Hadwiger's conjecture and topological bounds

Abstract: The Odd Hadwiger's conjecture, formulated by Gerards and Seymour in 1995, is a substantial strengthening of Hadwiger's famous coloring conjecture from 1943. We investigate whether the hierarchy of topological lower bounds on the chromatic number, introduced by Matou\v{s}ek and Ziegler (2003) and refined recently by Daneshpajouh and Meunier (2023), forms a potential avenue to a disproof of Hadwiger's conjecture or its odd-minor variant. In this direction, we prove that, in a very general sense, every graph $G$ that admits a topological lower bound of $t$ on its chromatic number, contains $K_{\lfloor t/2\rfloor +1}$ as an odd-minor. This solves a problem posed by Simonyi and Zsb\'{a}n [European Journal of Combinatorics, 31(8), 2110--2119 (2010)]. We also prove that if for a graph $G$ the Dol'nikov-K\v{r}\'{i}\v{z} lower bound on the chromatic number (one of the lower bounds in the aforementioned hierarchy) attains a value of at least $t$, then $G$ contains $K_t$ as a minor. Finally, extending results by Simonyi and Zsb\'{a}n, we show that the Odd Hadwiger's conjecture holds for Schrijver and Kneser graphs for any choice of the parameters. The latter are canonical examples of graphs for which topological lower bounds on the chromatic number are tight.