18 research outputs found
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
Resolution of a conjecture about linking ring structures
An LR-structure is a tetravalent vertex-transitive graph together with a
special type of a decomposition of its edge-set into cycles. LR-structures were
introduced in a paper by P. Poto\v{c}nik and S. Wilson, titled `Linking rings
structures and tetravalent semisymmetric graphs', in Ars Math. Contemp. 7
(2014), as a tool to study tetravalent semisymmetric graphs of girth 4. In this
paper, we use the methods of group amalgams to resolve some problems left open
in the above-mentioned paper
Recent trends and future directions in vertex-transitive graphs
A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade