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Nowhere-zero 5-flow on signed ladders

Leila Parsaei-Majd

TL;DR

The paper addresses Bouchet's conjecture, which posits that every flow-admissible signed graph has a nowhere-zero $6$-flow, by focusing on circular and Möbius ladders. It proves that signed circular ladders $CL_n$ (for $n\ge5$) and signed Möbius ladders $ML_n$ admit nowhere-zero $5$-flows, with a single exceptional $CL_4$ signature that lacks a $5$-flow but possesses a $6$-flow, using switching equivalence, $1$-factorizations, and constructive extensions via positive-square subgraphs. The results rely on base-case analyses ($CL_5$, $CL_6$, $ML_4$, $ML_5$), plus inductive arguments on negative-edge configurations to extend flows to larger ladders, including careful handling of exceptional cases through explicit patterns. Together, they strengthen Bouchet's conjecture for these ladder families and provide explicit, constructive flow methods with potential extensions to broader ladder-like graphs, such as generalized Petersen graphs.

Abstract

In 1983, Bouchet conjectured that every flow-admissible signed graph admits a nowhere-zero $6$-flow. In this paper, we prove that Bouchet's conjecture holds for all signed ladders, circular and Möbius ladders. In fact, all signed ladders, circular and Möbius ladders admit a nowhere-zero $5$-flow except for one case of signed circular ladders. Of course, the exception also has a nowhere-zero $6$-flow.

Nowhere-zero 5-flow on signed ladders

TL;DR

The paper addresses Bouchet's conjecture, which posits that every flow-admissible signed graph has a nowhere-zero -flow, by focusing on circular and Möbius ladders. It proves that signed circular ladders (for ) and signed Möbius ladders admit nowhere-zero -flows, with a single exceptional signature that lacks a -flow but possesses a -flow, using switching equivalence, -factorizations, and constructive extensions via positive-square subgraphs. The results rely on base-case analyses (, , , ), plus inductive arguments on negative-edge configurations to extend flows to larger ladders, including careful handling of exceptional cases through explicit patterns. Together, they strengthen Bouchet's conjecture for these ladder families and provide explicit, constructive flow methods with potential extensions to broader ladder-like graphs, such as generalized Petersen graphs.

Abstract

In 1983, Bouchet conjectured that every flow-admissible signed graph admits a nowhere-zero -flow. In this paper, we prove that Bouchet's conjecture holds for all signed ladders, circular and Möbius ladders. In fact, all signed ladders, circular and Möbius ladders admit a nowhere-zero -flow except for one case of signed circular ladders. Of course, the exception also has a nowhere-zero -flow.

Paper Structure

This paper contains 3 sections, 10 theorems, 2 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Nowhere-zero flow on signed circular ladder $CL_{n}$
  3. Nowhere-zero flow on signed Möbius ladder $ML_{n}$

Key Result

Theorem 2.1

Let $G$ be a flow-admissible signed cubic graph with two negative edges. If $G$ is bipartite, then it has a nowhere-zero $k$-flow with $k\leqslant 4$.

Figures (15)

  • Figure 1:
  • Figure 2:
  • Figure 3: $CL_{5}$ with two and three negative edges
  • Figure 4: $CL_{6}$ with three and four negative edges
  • Figure 5: A subladder $H$
  • ...and 10 more figures

Theorems & Definitions (22)

  • Theorem 2.1: twonegative
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 12 more