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As a strengthening of Hadwiger's conjecture, Gerards and Seymour conjecturedthat every graph with no odd $K_t$ minor is $(t-1)$-colorable. We prove twoweaker variants of this conjecture. Firstly, we show that for each $t \geq 2$,every graph with no odd $K_t$ minor has a partition of its vertex set into$6t-9$ sets $V_1, \dots, V_{6t-9}$ such that each $V_i$ induces a subgraph ofbounded maximum degree. Secondly, we prove that for each $t \geq 2$, everygraph with no odd $K_t$ minor has a partition of its vertex set into $10t-13$sets $V_1, \dots, V_{10t-13}$ such that each $V_i$ induces a subgraph withcomponents of bounded size. The second theorem improves a result ofKawarabayashi (2008), which states that the vertex set can be partitioned into$496t$ such sets.

Improper coloring of graphs with no odd clique minor