Frustration indices of signed subcubic graphs

TL;DR

This work studies the frustration index $F(G,\sigma)$ of signed subcubic graphs, a signed-graph reformulation of Max-Cut. It establishes a sharp bound $F(G,\sigma) \le \frac{1}{3}v(G)$ for signed $2$-edge-connected simple subcubic graphs, with five exceptional graphs and an infinite family attaining equality, and a tightened bound $F(G,\sigma) \le \frac{3v(G)+2}{8}$ for graphs with cut-edges, with a full characterization of equality cases. It extends these results to corollaries for cubic graphs, notably yielding $F(G,\sigma) \le \frac{2}{9}e(G)$ for large classes and demonstrating tightness via infinite families. The proofs combine a minimum counterexample framework, detailed block-leaf analysis, and a crucial inequality $|X_3|+|Y_0| \le |Y_2|$ derived from a careful vertex-partition argument, advancing understanding of frustration indices in restricted graph classes and informing potential tighter girth-based conjectures.

Abstract

The frustration index of a signed graph is defined as the minimum number of negative edges among all switching-equivalent signatures. This can be regarded as a generalization of the classical \textsc{Max-Cut} problem in graphs, as the \textsc{Max-Cut} problem is equivalent to determining the frustration index of signed graphs with all edges being negative signs. In this paper, we prove that the frustration index of an $n$-vertex signed connected simple subcubic graph, other than $(K_4, -)$, is at most $\frac{3n + 2}{8}$, and we characterize the family of signed graphs for which this bound is attained. This bound can be further improved to $\frac{n}{3}$ for signed $2$-edge-connected simple subcubic graphs, with the exceptional signed graphs being characterized. As a corollary, every signed $2$-edge-connected simple cubic graph on at least $10$ vertices and with $m$ edges has its frustration index at most $\frac{2}{9}m$, where the upper bound is tight as it is achieved by an infinite family of signed cubic graphs.

Figures (10)

• Figure 1: Five exceptional signed subcubic graphs with $F(G, \sigma)>\frac{1}{3}v(G)$
• Figure 2: A subgraph $\widehat{G_i}$
• Figure 3: Examples illustrating the tightness of \ref{['thm:1ec']}
• Figure 4: Two $8$-vertex cubic graphs $W_1$ and $W_2$
• Figure 5: Configurations in \ref{['lem:34']}

Theorems & Definitions (39)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Conjecture 1.4
• Lemma 2.2
• Lemma 2.3
• proof : Proof of \ref{['thm:main']}
• Proposition 2.4
• proof
• proof : Proof of the first part of \ref{['thm:1ec']}