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Bouchet's conjecture for cyclically 5-edge-connected, cubic signed graphs

Kathryn Nurse

TL;DR

Bouchet's conjecture posits that every flow-admissible signed graph admits a nowhere-zero $6$-flow. The paper proves this conjecture for the class of cyclically $5$-edge-connected, cubic signed graphs by combining a DeVos et al. $\mathbb{Z}_6$-flow framework with a matching-based augmentation to a $6$-flow. The method enforces edge values in $\{1,2,3,4,5\}$ and uses boundary conditions to create sources and near-sources, then leverages a parity-enhanced $\mathbb{Z}_2 \times \mathbb{Z}_3$-flow to obtain a suitable $\mathbb{Z}_6$-flow with an even number of sources. A perfect matching in the induced $6$-regular multigraph is found (via Tutte's theorem) and edges in the matching are reversed to yield a nowhere-zero $6$-flow on the original graph, completing the proof. The result advances Bouchet's conjecture under strong connectivity and suggests potential reductions of the connectivity assumptions in future work.

Abstract

A 1983 conjecture of Bouchet states that every flow-admissible signed graph has a nowhere-zero six-flow. We prove this conjecture for cyclically five-edge-connected, cubic signed graphs.

Bouchet's conjecture for cyclically 5-edge-connected, cubic signed graphs

TL;DR

Bouchet's conjecture posits that every flow-admissible signed graph admits a nowhere-zero -flow. The paper proves this conjecture for the class of cyclically -edge-connected, cubic signed graphs by combining a DeVos et al. -flow framework with a matching-based augmentation to a -flow. The method enforces edge values in and uses boundary conditions to create sources and near-sources, then leverages a parity-enhanced -flow to obtain a suitable -flow with an even number of sources. A perfect matching in the induced -regular multigraph is found (via Tutte's theorem) and edges in the matching are reversed to yield a nowhere-zero -flow on the original graph, completing the proof. The result advances Bouchet's conjecture under strong connectivity and suggests potential reductions of the connectivity assumptions in future work.

Abstract

A 1983 conjecture of Bouchet states that every flow-admissible signed graph has a nowhere-zero six-flow. We prove this conjecture for cyclically five-edge-connected, cubic signed graphs.
Paper Structure (3 sections, 6 theorems, 2 equations, 2 figures)

This paper contains 3 sections, 6 theorems, 2 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basics
  3. Proof of main result

Key Result

Theorem 1.2

Every flow-admissible, cyclically 5-edge-connected, cubic signed graph has a nowhere-zero 6-flow.

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (12)

  • Conjecture 1.1: Bouchet Bouchet
  • Theorem 1.2
  • Lemma 2.2
  • proof
  • Theorem 2.3: folklore
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2: DLLZZ
  • Lemma 3.3
  • ...and 2 more