2025/12/19 by Chen, Lei, Praeger, Cheryl
#05C25 #20B25 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2512.17244
We extend the notion of an H-normal quotient digraph of an H-vertex-transitive digraph to that of an H-subnormal quotient digraph. Using these concepts, together with bipartite halves of bipartite digraphs, we show that, for each finite connected H-vertex-transitive, (H,s)-arc-transitive digraph with s\geqslant6, either some H-normal quotient is a directed cycle of length at least 3, or there is an (L,t)-arc-transitive digraph with t\geqslant (s-3)/2, and L a vertex-quasiprimitive almost simple group with socle a composition factor of H. This connection demonstrates that, to understand finite s-arc-transitive digraphs with large s, those admitting a vertex-quasiprimitive almost simple s-arc-transitive subgroup of automorphisms play a central role. We show that for each s and each odd valency k, there are infinitely many (H,s)-arc-transitive digraphs of valency k with H a finite alternating group. In addition we discovered a novel construction which takes as input a connected non-bipartite H-vertex-transitive, (H,s)-arc-transitive digraph, and outputs a connected bipartite G-vertex-transitive, (G,2s)-arc-transitive digraph with G=(H× H).2. This leads to construction of vertex-bi-quasiprimitive s-arc-transitive digraphs, for arbitrarily large s. Our investigations yield several new open problems.