Bicircular matroids are 3-colorable

Hugo Hadwiger proved that a graph that is not 3-colorable must have a K4minor and conjectured that a graph that is not k-colorable must have a Kk+1minor. By using the Hochstättler-Nešetřil definition for the chromatic number of an oriented matroid, we formulate a generalized version of Hadwiger’s conjecture that might hold for the class of oriented matroids. In particular, it is possible that every oriented matroid with no M(K4)-minor is 3-colorable. The fact that K4-minor-free graphs are characterized as series-parallel networks leads to an easy proof that they are all 3-colorable. We show how to extend this argument to a particular subclass of M(K4)-minor-free oriented matroids. Specifically we generalize the notion of being series-parallel to oriented matroids, and then show that generalized series-parallel oriented matroids are 3-colorable. To illustrate the method, we show that every orientation of a bicircular matroid is 3-colorable.