7 research outputs found
A characterization of metacirculants
AbstractMetacirculants were introduced by Alspach and Parsons in 1982 and have been a rich source of various topics since then, including the Hamiltonian path problem in metacirculants. A metacirculant has a vertex-transitive metacyclic subgroup of automorphisms, and a long-standing interesting question in the area is if the converse statement is true, namely, whether a graph with a vertex-transitive metacyclic automorphism group is a metacirculant. We shall answer this question in the negative, and then present a classification of cubic metacirculants
On tetravalent half-arc-transitive graphs of girth 5
A subgroup of the automorphism group of a graph \G is said to be {\em
half-arc-transitive} on \G if its action on \G is transitive on the vertex
set of \G and on the edge set of \G but not on the arc set of \G.
Tetravalent graphs of girths and admitting a half-arc-transitive group
of automorphisms have previously been characterized. In this paper we study the
examples of girth . We show that, with two exceptions, all such graphs only
have directed -cycles with respect to the corresponding induced orientation
of the edges. Moreover, we analyze the examples with directed -cycles, study
some of their graph theoretic properties and prove that the -cycles of such
graphs are always consistent cycles for the given half-arc-transitive group. We
also provide infinite families of examples, classify the tetravalent graphs of
girth admitting a half-arc-transitive group of automorphisms relative to
which they are tightly-attached and classify the tetravalent
half-arc-transitive weak metacirculants of girth