Abstract
We analyse the normal quotient structure of several infinite families of finite connected edge-transitive, four-valent oriented graphs. These families were singled out by Marusic and others to illustrate various different internal structures for these graphs in terms of their alternating cycles (cycles in which consecutive edges have opposite orientations). Studying the normal quotients gives fresh insights into these oriented graphs: in particular we discovered some unexpected ‘cross-overs’ between these graph families when we formed normal quotients. We determine which of these oriented graphs are ‘basic’, in the sense that their only proper normal quotients are degenerate. Moreover, we show that the three types of edge-orientations studied are the only orientations, of the underlying undirected graphs in these families, which are invariant under a group action which is both vertex-transitive and edge-transitive.
Highlights
The graphs we study are simple, connected, undirected graphs of valency four, admitting an orientation of their edges preserved by a vertex-transitive and edge-transitive subgroup of the automorphism group, that is, graphs of valency four admitting a half-arc-transitive group action
These families are based on one of two infinite families of underlying unoriented valency four graphs (called X(r) and Y (r) for positive integers r), and for each family three different edge-orientations are induced by three different subgroups of their automorphism groups (Section 1.3); the three different edge-orientations correspond to the three different attachment properties of their alternating cycles
In this paper it is shown in Theorem 1.1 that these are essentially the only edge-orientations of the underlying graphs which are invariant under a half-arc-transitive group action
Summary
The graphs we study are simple, connected, undirected graphs of valency four, admitting an orientation of their edges preserved by a vertex-transitive and edge-transitive subgroup of the automorphism group, that is, graphs of valency four admitting a half-arc-transitive group action. The families of oriented graphs studied in this paper were singled out in [9, 10] because they demonstrate three different extremes for the structure of their alternating cycles: namely the alternating cycles are ‘loosely attached’, ‘antipodally attached’ or ‘tightly attached’ (see Subsection 1.2) These families are based on one of two infinite families of underlying unoriented valency four graphs (called X(r) and Y (r) for positive integers r), and for each family three different edge-orientations are induced by three different subgroups of their automorphism groups (Section 1.3); the three different edge-orientations correspond to the three different attachment properties of their alternating cycles. In addition we determine in Theorem 1.2 those oriented graphs in these families which are ‘basic’ in the sense that all their proper normal quotients are degenerate (see Subsection 1.1)
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