Strong colouring of matroid-graph pairs
Let G=(V,E) be a graph with maximum degree [math]\Delta \geq 2[/math], and let M=(V,r) be a matroid that has [math]2 \Delta[/math] disjoint bases. Is it true that M has [math]2 \Delta[/math] disjoint bases that are all independent in G?
Aharoni and Berger  proved that if the condition holds, then M has a G-independent base. This was improved by Aharoni, Berger, and Sprüssel : if [math]\Delta \geq 3[/math] and the condition holds, then M has two disjoint G-independent bases (interestingly, the case [math]\Delta =2[/math] is open).