Tetravalent Graphs Admitting Half-Transitive Group Actions: Alternating Cycles

Tetravalent Graphs Admitting Half-Transitive Group Actions: Alternating Cycles

In this paper we study finite, connected, 4-valent graphs X which admit an action of a group G which is transitive on vertices and edges, but not transitive on the arcs of X. Such a graph X is said to be (G, 1 2)-transitive. The group G induces an orientation of the edges of X, and a certain class of cycles of X (called alternating cycles) determined by the group G is identified as having an im...

Tetravalent half-edge-transitive graphs and non-normal Cayley graphs

Half-Transitive Group Actions on Finite Graphs of Valency 4

The action of a subgroup G of automorphisms of a graph X is said to be 2 -transitive if it is vertexand edgebut not arc-transitive. In this case the graph X is said to be (G, 2)-transitive. In particular, X is 1 2 -transitive if it is (Aut X, 1 2)-transitive. The 2 -transitive action of G on X induces an orientation of the edges of X which is preserved by G. Let X have valency 4. An even length...

Tetravalent half-arc-transitive graphs of order p4

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Let p and q be primes. It is known that no tetravalent half-arc-transitive graphs of order 2p2 exist and a tetravalent half-arctransitive graph of order 4p must be non-Cayley; such a non-Cayley graph exists if and only if p − 1 is divisible by 8 and it is unique for a given o...

Finite Graphs of Valency 4 and Girth 4 Admitting Half - Transitive Group

Finite graphs of valency 4 and girth 4 admitting 1 2-transitive group actions, that is vertex-and edge-but not arc-transitive group actions, are investigated. A graph is said to be 1 2-transitive if its automorphism group acts 1 2-transitively. There is a natural (balanced) orientation of the edge set of a 1 2-transitive graph induced and preserved by its auto-morphism group. Among others it is...