Short rainbow circuit conjecture for regular matroids

Let MM be a simple regular matroid, let r(M)r(M) denote its rank, and let cc be an (r(M)+1)(r(M)+1)-colouring of MM. A rainbow circuit is a circuit whose elements have pairwise distinct colours. It is short when

Cr(M)+22.|C|\leq \left\lfloor \frac{r(M)+2}{2}\right\rfloor.

Short rainbow circuit conjecture. If (M,c)(M,c) has no colour-singular elements, then MM has a short rainbow circuit. This conjecture extends the corresponding rank-two case from graphs to regular matroids. The supplied text does not state whether it has been resolved.

Sources & referencesView supporting material

Primary source

Sean McGuinness, “Short Rainbow Circuits in Regular Matroids”, arXiv:2601.17624 (2026).

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