Cyclically $5$-edge-connected snarks with resistance $2$ and flow resistance $n$
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| Format: | Preprint |
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2026
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| _version_ | 1866908991046549504 |
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| author | Mattiolo, Davide Negrini, Pietro Pagani, Silvia M. C. |
| author_facet | Mattiolo, Davide Negrini, Pietro Pagani, Silvia M. C. |
| contents | Snarks are $2$-connected cubic graphs that do not admit a proper $3$-edge-coloring. For a cubic graph $G$, its resistance $r(G)$ is the minimum number of edges whose removal results in a $3$-edge-colorable graph, while its flow resistance $r_f(G)$ is the minimum number of edges whose removal results in a graph admitting a nowhere-zero $\mathbb{Z}_2 \times \mathbb{Z}_2$-flow. In this paper, we provide an affirmative answer to a question recently posed by Allie, Máčajová, and Škoviera by constructing a family of cyclically $5$-edge-connected snarks for which the ratio $r_f(G)/r(G)$ is arbitrarily large. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_22501 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cyclically $5$-edge-connected snarks with resistance $2$ and flow resistance $n$ Mattiolo, Davide Negrini, Pietro Pagani, Silvia M. C. Combinatorics 05C15, 05C21, 05C40 Snarks are $2$-connected cubic graphs that do not admit a proper $3$-edge-coloring. For a cubic graph $G$, its resistance $r(G)$ is the minimum number of edges whose removal results in a $3$-edge-colorable graph, while its flow resistance $r_f(G)$ is the minimum number of edges whose removal results in a graph admitting a nowhere-zero $\mathbb{Z}_2 \times \mathbb{Z}_2$-flow. In this paper, we provide an affirmative answer to a question recently posed by Allie, Máčajová, and Škoviera by constructing a family of cyclically $5$-edge-connected snarks for which the ratio $r_f(G)/r(G)$ is arbitrarily large. |
| title | Cyclically $5$-edge-connected snarks with resistance $2$ and flow resistance $n$ |
| topic | Combinatorics 05C15, 05C21, 05C40 |
| url | https://arxiv.org/abs/2604.22501 |