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

Abstract

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 K 2 (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.