Frustration indices of signed subcubic graphs

TL;DR

This paper establishes upper bounds on the frustration index of signed subcubic graphs, characterizes extremal cases, and relates the frustration index to graph connectivity and size, extending classical Max-Cut concepts.

Contribution

It provides new bounds for the frustration index of signed subcubic graphs, characterizes cases where bounds are tight, and links frustration index to graph connectivity and size.

Findings

Frustration index of signed connected subcubic graphs is at most (3n+2)/8.

Bound improves to n/3 for 2-edge-connected signed subcubic graphs.

Frustration index of signed cubic graphs with ≥10 vertices is at most (2/9) times the number of edges.

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.