Two Disjoint Independent Bases in Matroid-Graph Pairs

Abstract

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