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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2511.01556 |
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| _version_ | 1866912685452427264 |
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| author | Mattiolo, Davide |
| author_facet | Mattiolo, Davide |
| contents | A set $R\subseteq E(G)$ of a graph $G$ is $k$-removable if $G-R$ has a nowhere-zero $k$-flow. We prove that every graph $G$ admitting a nowhere-zero $4$-flow has a $3$-removable subset consisting of at most $\frac{1}{6}|E(G)|$ edges. This gives a positive answer to a conjecture of M. DeVos, J. McDonald, I. Pivotto, E. Rollová and R. Šámal [$3$-Flows with large support, J. Comb. Theory Ser. B 144 (2020), 32-80] in the case of graphs admitting a nowhere-zero $4$-flow.
Moreover, Hoffmann-Ostenhof recently conjectured that every cubic graph with a nowhere-zero $4$-flow has a $4$-removable edge. Bipartite cubic graphs verify this conjecture. Our result gives an approximation for Hoffmann-Ostenhof's Conjecture in the non-bipartite case.
Finally, for cubic graphs, our result implies that every $3$-edge-colorable cubic graph $G$ contains a subgraph $H$ whose connected components are either cycles or subdivisions of bipartite cubic graphs, such that $|E(H)|\ge \frac{5}{6}|E(G)|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01556 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On removable edge subsets in graphs with a nowhere-zero $4$-flow Mattiolo, Davide Combinatorics 05C21, 05C70 A set $R\subseteq E(G)$ of a graph $G$ is $k$-removable if $G-R$ has a nowhere-zero $k$-flow. We prove that every graph $G$ admitting a nowhere-zero $4$-flow has a $3$-removable subset consisting of at most $\frac{1}{6}|E(G)|$ edges. This gives a positive answer to a conjecture of M. DeVos, J. McDonald, I. Pivotto, E. Rollová and R. Šámal [$3$-Flows with large support, J. Comb. Theory Ser. B 144 (2020), 32-80] in the case of graphs admitting a nowhere-zero $4$-flow. Moreover, Hoffmann-Ostenhof recently conjectured that every cubic graph with a nowhere-zero $4$-flow has a $4$-removable edge. Bipartite cubic graphs verify this conjecture. Our result gives an approximation for Hoffmann-Ostenhof's Conjecture in the non-bipartite case. Finally, for cubic graphs, our result implies that every $3$-edge-colorable cubic graph $G$ contains a subgraph $H$ whose connected components are either cycles or subdivisions of bipartite cubic graphs, such that $|E(H)|\ge \frac{5}{6}|E(G)|$. |
| title | On removable edge subsets in graphs with a nowhere-zero $4$-flow |
| topic | Combinatorics 05C21, 05C70 |
| url | https://arxiv.org/abs/2511.01556 |