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arXiv · 2511.15226

Frustration indices of signed subcubic graphs

Sirui Chen·Jiaao Li·Zhouningxin Wang
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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.

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Sirui Chen, Jiaao Li, Zhouningxin Wang. 2025-11-19. Frustration indices of signed subcubic graphs. https://arxiv.org/abs/2511.15226

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