📝 SummarXiv

Abstract

We investigate spanning 1-factorizations in vertex-transitive digraphs of arbitrary out-degree d. The guiding question is whether every such digraph admits a spanning 1-factorization that includes, for each vertex v, all d out-edges (v, F1(v)) from v. Equivalently, does there always exist a spanning factorization containing the empty word together with the full set of generators {F1, ..., Fd}? This paper addresses 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 and F2. We show that certain block obstructions can prevent their simultaneous inclusion, while sharply transitive sets (and hence spanning 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.