Gerards–Seymour odd Hadwiger conjecture

Let $G$ be a graph. We say that $G$ has $K_t$ as an odd minor if there exist vertex-disjoint trees $T_1,\dots,T_t\subseteq G$ and functions

$$\chi_i:V(T_i)\to\{\text{black},\text{white}\}$$

for which each $\chi_i$ is a proper $2$-coloring of $T_i$, and for every $i\ne j$ there are vertices $x_i\in V(T_i)$ and $x_j\in V(T_j)$ with $\{x_i,x_j\}\in E(G)$ and $\chi_i(x_i)=\chi_j(x_j)$.

Odd Hadwiger conjecture. If

$$\chi(G)\geq t,$$

then $G$ has $K_t$ as an odd minor. Equivalently, if $G$ does not have $K_t$ as an odd minor, then

$$\chi(G)<t.$$

Gerards and Seymour proposed this parity-sensitive strengthening of Hadwiger's conjecture. The paper disproves it: for every fixed $\delta>0$ and all sufficiently large $t$, there is a graph with $\chi(G)\geq(\frac32-\delta)t$ having no $K_t$ odd minor.