Half-arc-transitive group actions with a small number of alternets

A graph X is said to be G-half-arc-transitive if G⩽Aut(X) acts transitively on the set of vertices of X and on the set of edges of X but does not act transitively on the set of arcs of X. Such graphs can be studied via corresponding alternets, that is, equivalence classes of the so-called reachability relation, first introduced by Cameron, Praeger and Wormald (1993) in [5]. If the vertex sets of two adjacent alternets either coincide or have half of their vertices in common the graph is said to be tightly attached. In this paper graphs admitting a half-arc-transitive group action with at most five alternets are considered. In particular, it is shown that if the number of alternets is at most three, then the graph is necessarily tightly attached, but there exist graphs with four and graphs with five alternets which are not tightly attached. The exceptional graphs all admit a partition giving the rose window graph R6(5,4) on 12 vertices as a quotient graph in case of four alternets, and a particular graph on 20 vertices in the case of five alternets.