Spanning Factorizations in Vertex-Transitive Digraphs of Degree 2

Abstract

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.