A new upper bound on the chromatic number of graphs with no odd $K_t$ minor

Abstract

Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. This is a strengthening of the famous Hadwiger's Conjecture. Geelen et al. proved that every graph with no odd $K_t$ minor is $O(t\sqrt{\log t})$-colorable. Using the methods the present authors and Postle recently developed for coloring graphs with no $K_t$ minor, we make the first improvement on this bound by showing that every graph with no odd $K_t$ minor is $O(t(\log t{)}^{\beta })$-colorable for every $\beta >1/4$.

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• Connectivity and choosability of graphs with no $K_t$ minor