Characterization of 2-arc-transitive partial cubes

Abstract

Let Γ be a graph with its automorphism group G and s(⩾0) an integer. We call a vertex sequence (v0,v1,…,vs) of length s+1 of Γ an s-arc if two consecutive vertices are adjacent and vi≠vi+2 for 0⩽i⩽s−2. Γ is called s-arc-transitive if G acts on its s-arc set transitively. If Γ is s-arc-transitive, but not (s+1)-arc-transitive, we call it an s-transitive graph. Γ is a partial cube if Γ can be embedded into a hypercube isometrically. In this paper, we characterized non-trivial regular 2-arc-transitive partial cubes as three classes of graphs: hypercubes, doubled Odd graphs and even cycles. As a corollary, we prove that there exist no regular s-transitive partial cubes for s⩾4 and characterized regular s-transitive partial cubes for s=2 or 3.