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Short Rainbow Circuits in Regular Matroids

Sean McGuinness

TL;DR

This work resolves a conjecture of DeVos et al. by proving that every simple regular matroid $M$ with a $(r(M)+1)$-colouring, where each colour class has at least two elements, contains short rainbow circuits. The authors develop a unifying framework based on SRC$4$-tuples—collections of four rainbow circuits with total size at most $2r(M)+4$ and each element lying in at most two circuits—and show this structure persists under the standard Seymour decomposition of regular matroids. A key strategy is to analyze circuit-achromatic colourings on binary matroids (via stratifications) and to prove the main result first for graphic and cographic matroids, then extend to all regular matroids through sums and extensions. The paper also handles extremal edge-count cases and duals thegraphic/cographic arguments to secure the general theorem. The result thus yields a robust short rainbow-circuit guarantee, with implications for rainbow structures in matroids and their decompositions, and it provides a constructive approach that underpins potential algorithmic applications.

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.

Short Rainbow Circuits in Regular Matroids

TL;DR

This work resolves a conjecture of DeVos et al. by proving that every simple regular matroid with a -colouring, where each colour class has at least two elements, contains short rainbow circuits. The authors develop a unifying framework based on SRC-tuples—collections of four rainbow circuits with total size at most and each element lying in at most two circuits—and show this structure persists under the standard Seymour decomposition of regular matroids. A key strategy is to analyze circuit-achromatic colourings on binary matroids (via stratifications) and to prove the main result first for graphic and cographic matroids, then extend to all regular matroids through sums and extensions. The paper also handles extremal edge-count cases and duals thegraphic/cographic arguments to secure the general theorem. The result thus yields a robust short rainbow-circuit guarantee, with implications for rainbow structures in matroids and their decompositions, and it provides a constructive approach that underpins potential algorithmic applications.

Abstract

DeVos et al conjectured that if is a simple, regular matroid and is a colouring of the elements of with colours, where each colour class has at least two elements, then contains a rainbow circuit of size at most We prove this conjecture by showing that for all such regular matroids there are four rainbow circuits for which and for which no element of belongs to more than two of the circuits.
Paper Structure (25 sections, 42 equations, 2 figures)

This paper contains 25 sections, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. A theorem for circuit-achromatic coloured matroids
  4. Colourings of Binary Matroids
  5. Proof for graphic and cographic matroids
  6. A lemma for graphic matroids
  7. The proof of Theorem \ref{['the-GraphicCographicMain']} for graphic matroids
  8. The proof of Theorem \ref{['the-GraphicCographicMain']} when $M$ is cographic
  9. Circuit-achromatic graphic matroids $M$ with $2r(M)-1$ elements
  10. Proof of Theorem \ref{['the-NoRainbowCircuit']} for $M = M(G)$ and $\varepsilon(M) = 2r(M)-1$
  11. Circuit-achromatic cographic matroids $M$ with $2r(M)-1$ elements
  12. Stratified Colourings in Co-Graphic Matroids
  13. The construction of $G$
  14. Completing the proof of Theorem \ref{['the-NoRainbowCircuit']} i) for $M$
  15. Circuit-achromatic graphic matroids $M$ with $2(r(M)-1)$ elements.
  16. ...and 10 more sections

Figures (2)

  • Figure 1: The two possible graphs for $G\simeq K_{23}^+.$
  • Figure 2: The operation $H \xrightarrow[v]{f_1, \dots ,f_k}H'$

Theorems & Definitions (79)

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  • ...and 69 more