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 $3$ and $4$ admitting a half-arc-transitive group of automorphisms have previously been characterized. In this paper we study the examples of girth $5$. We show that, with two exceptions, all such graphs only have directed $5$-cycles with respect to the corresponding induced orientation of the edges. Moreover, we analyze the examples with directed $5$-cycles, study some of their graph theoretic properties and prove that the $5$-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 $5$ 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 $5$

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

There exists no tetravalent half-arc-transitive graph of order 2p2

AbstractA graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not arc set. In this paper, we show that there is no tetravalent half-arc-transitive graph of order 2p2