Note on coloring graphs without odd--minors

We give a short proof that every graph G without an odd K k -minor is O ( k log k ) -colorable. This was first proved by Geelen et al. [J. Geelen, B. Gerards, B. Reed, P. Seymour, A. Vetta, On the odd-minor variant of Hadwigerʹs conjecture, J. Combin. Theory Ser. B 99 (1) (2009) 20–29]. We give a considerably simpler and shorter proof following their approach.