New structural results on tetravalent half-arc-transitive graphs

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Tetravalent graphs admitting a half-arc-transitive group of automorphisms, that is a subgroup of the automorphism group acting transitively on its vertices and its edges but not on its arcs, are investigated. One of the most fruitful approaches for the study of structural properties of such graphs is the well-known concept of alternating cycles and their intersections which was introduced by Marušič 20 years ago.In this paper a new parameter for such graphs, giving a further insight into their structure, is introduced. Various properties of this parameter are given and the parameter is completely determined for the tightly attached examples in which any two non-disjoint alternating cycles meet in half of their vertices. Moreover, the obtained results are used to establish a link between two frameworks for a systematic study of all tetravalent graphs admitting a half-arc-transitive group of automorphisms, namely the one proposed by Marušič and Praeger in 1999, and the much more recent one proposed by Al-bar, Al-kenai, Muthana, Praeger and Spiga which is based on the normal quotients method.New results on the graph of alternating cycles of a tetravalent graph admitting a half-arc-transitive group of automorphisms are obtained. A considerable step towards the complete answer to the question of whether the attachment number necessarily divides the radius in tetravalent half-arc-transitive graphs is made.A number of questions and open problems are posed.