Vertex-transitive expansions of (1, 3)-trees

A nonidentity automorphism of a graph is said to be semiregular   if all of its orbits are of the same length. Given a graph XX with a semiregular automorphism γγ, the quotient of XX relative to γγ is the multigraph X/γX/γ whose vertices are the orbits of γγ and two vertices are adjacent by an edge with multiplicity rr if every vertex of one orbit is adjacent to rr vertices of the other orbit. We say that XX is an expansion   of X/γX/γ. In [J.D. Horton, I.Z. Bouwer, Symmetric YY-graphs and HH-graphs, J. Combin. Theory Ser. B 53 (1991) 114–129], Horton and Bouwer considered a restricted sort of expansions (which we will call ‘strong’ in this paper) where every leaf of X/γX/γ expands to a single cycle in XX. They determined all cubic arc-transitive strong expansions of simple (1, 3)-trees, that is, trees with all of their vertices having valency 1 or 3, thus extending the classical result of Frucht, Graver and Watkins (see [R. Frucht, J.E. Graver, M.E. Watkins, The groups of the generalized Petersen graphs, Proc. Cambridge Philos. Soc. 70 (1971) 211–218]) about arc-transitive strong expansions of K2K2 (also known as the generalized Petersen graphs). In this paper another step is taken further by considering the possible structure of cubic vertex-transitive expansions of general (1,3)-multitrees (where vertices with double edges are also allowed); thus the restriction on every leaf to be expanded to a single cycle is dropped.