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arXiv · 2601.17624

Short Rainbow Circuits in Regular Matroids

Sean McGuinness
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Abstract

DeVos et al conjectured that if $M$ is a simple, regular matroid and $c$ is a colouring of the elements of $M$ with $r(M)+1$ colours, where each colour class has at least two elements, then $M$ contains a rainbow circuit of size at most $\lceil \frac {r(M)+1}2 \rceil.$ We prove this conjecture by showing that for all such regular matroids there are four rainbow circuits $C_i,\ i = 1,2,3,4$ for which $\sum_i |C_i| \le 2r(M) +4$ and for which no element of $M$ belongs to more than two of the circuits.

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Sean McGuinness. 2026-01-24. Short Rainbow Circuits in Regular Matroids. https://arxiv.org/abs/2601.17624

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