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Nowhere-zero 8-flows in 3-edge-connected signed graphs

Matt DeVos, Kathryn Nurse, Robert Šámal

TL;DR

The paper proves that every flow-admissible signed graph that is 3-edge-connected admits a nowhere-zero $8$-flow, advancing Bouchet's conjecture in the signed-graph setting by weakening previous connectivity assumptions. The authors reduce the problem to cubic, highly connected graphs, develop a Cycle Selection Algorithm to generate a disjoint collection of cycles with carefully classified types, and then construct a $ ext{Z}_3$-preflow aligned with these cycles. They upgrade the preflow to a full integer flow using auxiliary-graph and matching techniques, and finally combine with a secondary flow to obtain an 8-flow, contradicting any minimal counterexample. This approach blends structural cycle decompositions, parity analyses, and established 6-flow results to push toward the broader goal of 6-flows in signed graphs.

Abstract

In 1983, A. Bouchet extended W.T. Tutte's notion of nowhere-zero flows to signed graphs, and conjectured that every flow-admissible signed graph has a nowhere-zero 6-flow. In this paper we prove that every flow-admissible signed graph that is 3-edge-connected has a nowhere-zero 8-flow. This is a continuation of a previous paper where we proved the same conclusion under stronger assumptions.

Nowhere-zero 8-flows in 3-edge-connected signed graphs

TL;DR

The paper proves that every flow-admissible signed graph that is 3-edge-connected admits a nowhere-zero -flow, advancing Bouchet's conjecture in the signed-graph setting by weakening previous connectivity assumptions. The authors reduce the problem to cubic, highly connected graphs, develop a Cycle Selection Algorithm to generate a disjoint collection of cycles with carefully classified types, and then construct a -preflow aligned with these cycles. They upgrade the preflow to a full integer flow using auxiliary-graph and matching techniques, and finally combine with a secondary flow to obtain an 8-flow, contradicting any minimal counterexample. This approach blends structural cycle decompositions, parity analyses, and established 6-flow results to push toward the broader goal of 6-flows in signed graphs.

Abstract

In 1983, A. Bouchet extended W.T. Tutte's notion of nowhere-zero flows to signed graphs, and conjectured that every flow-admissible signed graph has a nowhere-zero 6-flow. In this paper we prove that every flow-admissible signed graph that is 3-edge-connected has a nowhere-zero 8-flow. This is a continuation of a previous paper where we proved the same conclusion under stronger assumptions.

Paper Structure

This paper contains 7 sections, 31 theorems, 3 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Reducing the main theorem
  3. Finding a usable cycle
  4. Using an unbalanced theta
  5. Constructing a sequence of generalized cycles using the Cycle Selection Algorithm
  6. Forming a ${\mathbb Z}_3$-preflow
  7. Proof of the main theorem

Key Result

Theorem 3

Every flow-admissible, 3-edge-connected signed graph has a nowhere-zero 8-flow.

Figures (14)

  • Figure 1: Orientations of signed edges. Positive edges on left, negative edges on right.
  • Figure 2: Uncontracting at $v$ with $\{e,f\}$.
  • Figure 3: A family of fish signed graphs: signed graphs having an equivalent signature with a single negative edge shown in red. The three black dots represent an even number (zero or more) of consecutive degree-two vertices.
  • Figure 4: A fish graph with partial orientation. The distinguished edge is unoriented and highlighted in grey. The signature has exactly one negative edge (with two opposite-pointing arrows).
  • Figure 5: A positive cycle $C_k$ with $|F_k| = 1$.
  • ...and 9 more figures

Theorems & Definitions (58)

  • Conjecture 1: Tutte
  • Conjecture 2: Bouchet
  • Theorem 3
  • Proposition 4: doi:10.1137/23M1615218
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 48 more