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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.13432 |
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| _version_ | 1866916966574325760 |
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| author | Faber, Vance |
| author_facet | Faber, Vance |
| contents | We investigate the existence of spanning 1-factorizations in vertex-transitive digraphs of out-degree d. The open question is whether every such digraph admits a spanning 1-factorization that includes, for each vertex v, all d out-edges (v,F_i(v)) from v.
This paper focuses on the case d=2. Using the structure of alternating cycles and block systems, we develop a block/phase framework that yields sufficient conditions for including both F_1,F_2. We show that certain block obstructions can prevent their simultaneous inclusion, while sharply transitive sets (and hence spanning 1-factorizations) always exist. Our results provide general constraints on feasible block sizes, describe the role of phase distributions, and illustrate the theory with concrete families, including coset digraphs on A_5. The necessity of the block criterion remains open, even in degree 2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13432 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spanning Factorizations in Vertex-Transitive Digraphs of Degree 2 Faber, Vance Combinatorics 05C20 We investigate the existence of spanning 1-factorizations in vertex-transitive digraphs of out-degree d. The open question is whether every such digraph admits a spanning 1-factorization that includes, for each vertex v, all d out-edges (v,F_i(v)) from v. This paper focuses on the case d=2. Using the structure of alternating cycles and block systems, we develop a block/phase framework that yields sufficient conditions for including both F_1,F_2. We show that certain block obstructions can prevent their simultaneous inclusion, while sharply transitive sets (and hence spanning 1-factorizations) always exist. Our results provide general constraints on feasible block sizes, describe the role of phase distributions, and illustrate the theory with concrete families, including coset digraphs on A_5. The necessity of the block criterion remains open, even in degree 2. |
| title | Spanning Factorizations in Vertex-Transitive Digraphs of Degree 2 |
| topic | Combinatorics 05C20 |
| url | https://arxiv.org/abs/2509.13432 |