Almost all 9-regular graphs have a modulo-5 orientation
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866915246524858368 |
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| author | Delcourt, Michelle Huq, Reaz Pralat, Pawel |
| author_facet | Delcourt, Michelle Huq, Reaz Pralat, Pawel |
| contents | In 1972 Tutte famously conjectured that every 4-edge-connected graph has a nowhere zero 3-flow; this is known to be equivalent to every 5-regular, 4-edge-connected graph having an edge orientation in which every in-degree is either 1 or 4. Jaeger conjectured a generalization of Tutte's conjecture, namely, that every $4p+1$-regular, $4p$-edge-connected graph has an edge orientation in which every in-degree is either $p$ or $3p+1$. Inspired by the work of Pralat and Wormald investigating $p=1$, for $p=2$ we show this holds asymptotically almost surely for random 9-regular graphs. It follows that the conjecture holds for almost all 9-regular, 8-edge-connected graphs. These results make use of the technical small subgraph conditioning method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_12103 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Almost all 9-regular graphs have a modulo-5 orientation Delcourt, Michelle Huq, Reaz Pralat, Pawel Combinatorics In 1972 Tutte famously conjectured that every 4-edge-connected graph has a nowhere zero 3-flow; this is known to be equivalent to every 5-regular, 4-edge-connected graph having an edge orientation in which every in-degree is either 1 or 4. Jaeger conjectured a generalization of Tutte's conjecture, namely, that every $4p+1$-regular, $4p$-edge-connected graph has an edge orientation in which every in-degree is either $p$ or $3p+1$. Inspired by the work of Pralat and Wormald investigating $p=1$, for $p=2$ we show this holds asymptotically almost surely for random 9-regular graphs. It follows that the conjecture holds for almost all 9-regular, 8-edge-connected graphs. These results make use of the technical small subgraph conditioning method. |
| title | Almost all 9-regular graphs have a modulo-5 orientation |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2210.12103 |