[Paper Review] Signed graphs with two negative edges

Edita Rollová, Michael Schubert+1 more|arXiv (Cornell University)|Apr 27, 2016

Advancements in Battery Materials6 references3 citations

TL;DR

This paper investigates the flow number of flow-admissible signed cubic graphs with exactly two negative edges, proving that such graphs admit a nowhere-zero 7-flow, and tighter bounds under additional structural conditions—such as bipartite or 3-edge-colorable underlying graphs—where the flow number is at most 4 or 6, respectively. The results support Bouchet’s 6-flow conjecture and link it to Tutte’s 5-flow conjecture.

ABSTRACT

The presented paper studies the flow number $F(G,σ)$ of flow-admissible signed graphs $(G,σ)$ with two negative edges. We restrict our study to cubic graphs, because for each non-cubic signed graph $(G,σ)$ there is a set ${\cal G}(G,σ)$ of cubic graphs such that $F(G, σ) \leq \min \{F(H,σ_H) : (H,σ_H) \in {\cal G}(G)\}$. We prove that $F(G,σ) \leq 6$ if $(G,σ)$ contains a bridge and $F(G,σ) \leq 7$ in general. We prove better bounds, if there is an element $(H,σ_H)$ of ${\cal G}(G,σ)$ which satisfies some additional conditions. In particular, if $H$ is bipartite, then $F(G,σ) \leq 4$ and the bound is tight. If $H$ is 3-edge-colorable or critical or if it has a sufficient cyclic edge-connectivity, then $F(G,σ) \leq 6$. Furthermore, if Tutte's 5-Flow Conjecture is true, then $(G,σ)$ admits a nowhere-zero 6-flow endowed with some strong properties.

Motivation & Objective

• To determine the maximum flow number of flow-admissible signed cubic graphs with exactly two negative edges.
• To investigate how structural properties of the underlying unsigned graph (e.g., bipartite, 3-edge-colorable, cyclic edge-connectivity) affect the flow number.
• To explore the implications of these results for Bouchet’s 6-flow conjecture and Tutte’s 5-flow conjecture.
• To establish tight bounds on the flow number under specific graph-theoretic conditions, such as bridge-freeness and edge-cut restrictions.
• To demonstrate that Tutte’s 5-flow conjecture implies Bouchet’s 6-flow conjecture for signed graphs with two negative edges.

Proposed method

• Reduction to cubic graphs via a construction set $\mathcal{G}(G,\sigma)$, ensuring $F(G,\sigma) \leq \min\{F(H,\sigma_H) : (H,\sigma_H) \in \mathcal{G}(G,\sigma)\}$.
• Application of flow decomposition and orientation techniques, including Kirchhoff’s law and switching equivalence to simplify signatures.
• Use of 2-edge-cut and 3-edge-cut analysis to rule out non-flow-admissible configurations, especially when negative edges are involved.
• Construction of a nowhere-zero 7-flow by combining flows on smaller graphs using Corollary 5.2 and Lemma 3.5, ensuring flow value consistency across cut edges.
• Leveraging known results on 5-flows in all-positive graphs (e.g., from Seymour and others) to derive implications for signed graphs.
• Employing cyclic edge-connectivity and oddness parameters to bound the flow number, particularly in the context of $G^*$, the unsigned graph derived from $(G,\sigma)$.

Experimental results

Research questions

• RQ1 What is the maximum flow number $F(G,\sigma)$ for a flow-admissible signed cubic graph with exactly two negative edges?
• RQ2 How do structural properties like bipartiteness, 3-edge-colorability, or cyclic edge-connectivity of the underlying graph affect the flow number?
• RQ3 Can Tutte’s 5-flow conjecture imply Bouchet’s 6-flow conjecture for signed graphs with two negative edges?
• RQ4 Under what conditions can the flow number be reduced below 7, and what are the tightest possible bounds?
• RQ5 How does the presence of bridges or small edge-cuts influence the existence and value of nowhere-zero flows?

Key findings

• The flow number $F(G,\sigma) \leq 6$ if the underlying unsigned graph $G^*$ is bipartite, and this bound is tight.
• If the underlying graph $G^*$ is 3-edge-colorable or critical, or has sufficient cyclic edge-connectivity, then $F(G,\sigma) \leq 6$.
• For bridgeless signed cubic graphs with two negative edges, $F(G,\sigma) \leq 7$, and this bound is tight in general.
• If Tutte’s 5-flow conjecture holds, then every signed cubic graph with two negative edges admits a nowhere-zero 6-flow with strong structural properties: positive flow values, flow value 1 on negative edges, and all 5-flows concentrated on a single path.
• The flow number of a signed graph with two negative edges is bounded above by the minimum flow number over a set of cubic graphs derived from it, ensuring the bound is preserved under reduction.
• No signed cubic graph with two negative edges admits a nowhere-zero 5-flow if it is not flow-admissible, and such graphs are ruled out by small edge-cuts involving negative edges.