Two Disjoint Independent Bases in Matroid-Graph Pairs

Abstract

A result of Haxell (Graphs Comb 11:245---248, 1995) is that if $$G$$G is a graph of maximal degree $$\Delta $$Δ, and its vertex set is partitioned into sets $$V_i$$Vi of size $$2\Delta $$2Δ, then there exists an independent system of representatives (ISR), namely an independent set in the graph consisting of one vertex from each $$V_i$$Vi. Aharoni and Berger (Trans Am Math Soc 358:4895---4917, 2006) generalized this result to matroids: if a matroid $$\mathcal {M}$$M and a graph $$G$$G with maximal degree $$\Delta $$Δ share the same vertex set, and if there exist $$2\Delta $$2Δ disjoint bases of $$\mathcal {M}$$M, then there exists a base of $$\mathcal {M}$$M that is independent in $$G$$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 $$2\Delta (G)$$2Δ(G) disjoint $$G$$G-independent bases. In this paper we make a modest step: proving that for $$\Delta \ge 3$$Δ?3, under this condition there exist two disjoint $$G$$G-independent bases. The proof is topological.