2025/09/16 by Vance Faber, Faber, Vance
Computer Science · Mathematics · #05C20 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Interconnection Networks and Systems #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2509.13432
openalex publication_date 2025/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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,Fi(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 F1,F2. 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 A5. The necessity of the block criterion remains open, even in degree 2.