Strong colouring of matroid-graph pairs

Definitions and mathematical statement

Let $G=(V,E)$ be a finite simple graph of maximum degree $\Delta\ge2$, and let $M=(V,\mathcal I)$ be a matroid on the same vertex set. Its independent sets are hereditary and satisfy exchange: if $I,J\in\mathcal I$ with $|I|<|J|$, some $e\in J\setminus I$ can be added to $I$ while remaining independent. The maximal independent sets, called bases, all have the same size $r$.

Assume that $M$ contains $2\Delta$ pairwise disjoint bases. Must it contain bases $B_1,\ldots,B_{2\Delta}$ such that \[ B_i\cap B_j=\varnothing\quad(i\ne j),\qquad E(G[B_i])=\varnothing\quad(1\le i\le2\Delta)? \] Thus every selected basis must also be a stable vertex set in the graph. The conclusion may use entirely different bases from those witnessing the initial packing assumption.

The entire ground set need not be exhausted by the selected bases: the hypothesis says that disjoint bases exist, not that $V$ is their union. Stability is imposed separately within each basis, so edges joining two different selected bases are permitted. The matroid is arbitrary and is not assumed to be a partition matroid or representable over a field. The same family must satisfy the graph restrictions and the basis-packing requirement simultaneously.

Short English statement

Does packing $2\Delta$ disjoint matroid bases suffice to pack that many bases that are each stable in a graph of maximum degree $\Delta$?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3000082 (AMR-029-0082), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Egres Open - List of open problems, Open-problem page: Strong colouring of matroid-graph pairs. Original source indexed by AMR-029-0082. https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems.
• S3 EGRES Open, Strong colouring of matroid-graph pairs. Individual source page and explanatory remarks. https://lemon.cs.elte.hu/egres/open/Strong_colouring_of_matroid-graph_pairs.