Two Disjoint Independent Bases in Matroid-Graph Pairs

A result of Haxell (Graphs Comb 11:245–248, 1995) is that if G is a graph of maximal degree Δ, and its vertex set is partitioned into sets Vi of size 2Δ, then there exists an independent system of representatives (ISR), namely an independent set in the graph consisting of one vertex from each Vi. Aharoni and Berger (Trans Am Math Soc 358:4895–4917, 2006) generalized this result to matroids: if a matroid M and a graph G with maximal degree Δ share the same vertex set, and if there exist 2Δ disjoint bases of M, then there exists a base of M that is independent in G. In the Haxell result the matroid is a partition matroid. In that case, a well known conjecture, the strong coloring conjecture, is that in fact there is a partition into ISRs. This conjecture extends to the matroidal case: under the conditions above there exist 2Δ(G) disjoint G-independent bases. In this paper we make a modest step: proving that for (Formula presented.), under this condition there exist two disjoint G-independent bases. The proof is topological.