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 X with a semiregular automorphism γ, the quotient of X relative to γ is the multigraph X/γ whose vertices are the orbits of γ and two vertices are adjacent by an edge with multiplicity r if every vertex of one orbit is adjacent to r vertices of the other orbit. We say that X is an expansion of X/γ. In J.D. Horton, I.Z. Bouwer, Symmetric Y-graphs and H-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/γ expands to a single cycle in X. 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 K₂ (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.