# Strong colouring of matroid-graph pairs

Let G=(V,E) be a graph with maximum degree $\Delta \geq 2$, and let M=(V,r) be a matroid that has $2 \Delta$ disjoint bases. Is it true that M has $2 \Delta$ disjoint bases that are all independent in G?
Aharoni and Berger [1] proved that if the condition holds, then M has a G-independent base. This was improved by Aharoni, Berger, and Sprüssel [2]: if $\Delta \geq 3$ and the condition holds, then M has two disjoint G-independent bases (interestingly, the case $\Delta =2$ is open).