Separating Matchings in Cubic Graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| author | Gutiérrez, Juan Gómez, Renzo |
| author_facet | Gutiérrez, Juan Gómez, Renzo |
| contents | We study separating matchings in graphs, that is, matchings whose removal increases the number of connected components, and focus on determining the maximum size of such a matching in a graph $G$, denoted by $\mathrm{mms}(G)$. We show that every subcubic graph admits a separating matching, except for exactly eight graphs, which allows us to focus on bounding $\mathrm{mms}(G)$ for cubic graphs. Our main results show that every cubic graph $G$ on $n$ vertices that admits a separating matching satisfies $\mathrm{mms}(G) \ge n/2 - 2$. For bipartite cubic graphs, assuming a conjecture of Funk, the problem reduces to a recursively defined class $\mathcal{F}$, for which we prove that $\mathrm{mms}(G) \ge n/2 - 1$, up to four exceptional graphs. In contrast, we show that every claw-free cubic graph satisfies $\mathrm{mms}(G) = n/2$. These results extend previous work on matching cuts and disconnected $2$-factors, and provide the first systematic study of maximum separating matchings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17226 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Separating Matchings in Cubic Graphs Gutiérrez, Juan Gómez, Renzo Combinatorics 05C70, 05C40, 05C75 F.2.2; G.2.2 We study separating matchings in graphs, that is, matchings whose removal increases the number of connected components, and focus on determining the maximum size of such a matching in a graph $G$, denoted by $\mathrm{mms}(G)$. We show that every subcubic graph admits a separating matching, except for exactly eight graphs, which allows us to focus on bounding $\mathrm{mms}(G)$ for cubic graphs. Our main results show that every cubic graph $G$ on $n$ vertices that admits a separating matching satisfies $\mathrm{mms}(G) \ge n/2 - 2$. For bipartite cubic graphs, assuming a conjecture of Funk, the problem reduces to a recursively defined class $\mathcal{F}$, for which we prove that $\mathrm{mms}(G) \ge n/2 - 1$, up to four exceptional graphs. In contrast, we show that every claw-free cubic graph satisfies $\mathrm{mms}(G) = n/2$. These results extend previous work on matching cuts and disconnected $2$-factors, and provide the first systematic study of maximum separating matchings. |
| title | Separating Matchings in Cubic Graphs |
| topic | Combinatorics 05C70, 05C40, 05C75 F.2.2; G.2.2 |
| url | https://arxiv.org/abs/2604.17226 |