Short Rainbow Circuits in Regular Matroids

TL;DR

This paper proves a conjecture about the existence of small rainbow circuits in regular matroids under certain coloring conditions, by demonstrating the existence of four such circuits with bounded total size and limited element overlap.

Contribution

It establishes the existence of four rainbow circuits with controlled total size and element sharing in regular matroids, confirming a conjecture by DeVos et al.

Findings

Existence of four rainbow circuits with total size at most 2r(M)+4

No element belongs to more than two of the four circuits

Confirms the conjecture by DeVos et al.

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.