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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.11883 |
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| _version_ | 1866915857235443712 |
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| author | Wang, Zhiqian |
| author_facet | Wang, Zhiqian |
| contents | A signed graph $(G,σ)$ is a graph $G$ with a signature $σ$ labeling each edge with a positive or negative sign. Two signatures of $G$ are switching equivalent if one is obtained from the other by changing the signs of all edges in an edge-cut. The frustration index of a signed graph $(G, σ)$ is the minimum number of negative edges among all signatures equivalent to $σ$. A signed graph is critically $k$-frustrated if it has frustration index $k$, and the removal of any edge decreases its frustration index. A critically $k$-frustrated signed graph is prime if it has no subdivided edge (including multiedge) and none of its subgraphs is the edge-disjoint union of critically frustrated signed graphs. Steffen and Naserasr et al. conjectured that for any positive integer $k$, there are finitely many prime critically $k$-frustrated signed graphs. The cases $k=1,2,3$ have been proved to be true recently by Cappello et al.. In this paper, we show that the conjecture holds when $k=4$ and $5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11883 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finiteness of non-decomposable critically 4 and 5-frustrated signed graphs Wang, Zhiqian Combinatorics A signed graph $(G,σ)$ is a graph $G$ with a signature $σ$ labeling each edge with a positive or negative sign. Two signatures of $G$ are switching equivalent if one is obtained from the other by changing the signs of all edges in an edge-cut. The frustration index of a signed graph $(G, σ)$ is the minimum number of negative edges among all signatures equivalent to $σ$. A signed graph is critically $k$-frustrated if it has frustration index $k$, and the removal of any edge decreases its frustration index. A critically $k$-frustrated signed graph is prime if it has no subdivided edge (including multiedge) and none of its subgraphs is the edge-disjoint union of critically frustrated signed graphs. Steffen and Naserasr et al. conjectured that for any positive integer $k$, there are finitely many prime critically $k$-frustrated signed graphs. The cases $k=1,2,3$ have been proved to be true recently by Cappello et al.. In this paper, we show that the conjecture holds when $k=4$ and $5$. |
| title | Finiteness of non-decomposable critically 4 and 5-frustrated signed graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.11883 |