COLORING THE INTERSECTION OF TWO MATROIDS

A result from Aharoni and the first author of this paper [Trans. Amer. Math. Soc. 358 (2006), pp. 4895–4917] states that for any two positive integers p, q, where p divides q, if a matroid M is p-colorable and a matroid N is q-colorable then M ∩ N is (p + q)-colorable. In this paper we show that the assumption that p divides q is in fact redundant, and we also prove that M ∩ N is even p + q list-colorable. The result uses topology and relies on a new parameter yielding a lower bound for the topological connectivity of the intersection of two matroids.