Semi‐transitive graphs

Abstract

In this paper, we first consider graphs allowing symmetry groups which act transitively on edges but not on darts (directed edges). We see that there are two ways in which this can happen and we introduce the terms bi-transitive and semi-transitive to describe them. We examine the elementary implications of each condition and consider families of examples; primary among these are the semi-transitive spider-graphs PS(k,N;r ) and MPS(k,N;r ). We show how a product operation can be used to produce larger graphs of each type from smaller ones. We introduce the alternet of a directed graph. This links the two conditions, for each alternet of a semi-transitive graph (if it has more than one) is a bi-transitive graph. We show how the alternets can be used to understand the structure of a semi-transitive graph, and that the action of the group on the set of alternets can be an interesting structure in its own right. We use alternets to define the attachment number of the graph, and the important special cases of tightly attached and loosely attached graphs. In the case of tightly attached graphs, we show an addressing scheme to describe the graph with coordinates. Finally, we use the addressing scheme to complete the classification of tightly attached semi-transitive graphs of degree 4 begun by Marus ic and Praeger. This classification shows that nearly all such graphs are spider-graphs. ß

On quartic half-arc-transitive metacirculants

“…The nature of the action of ϕ will contradict half-arctransitivity of X, which thus proves that (AutX) v ∼ = Z 2 , as claimed. Note that (26) and (33) imply that r 2 = 7 u To see that ϕ is in fact a permutation of V (X), we only need to observe that it is injective. Consider the vertices of Table 2, which are of the form u Table 2 for each i = 1, 2, 3, and that the corresponding congruencies modulo 8 of their superscripts are again the eight possibilities 0, 1, .…”

“…With some of the research emphasis shifting to questions concerning structural properties of half-arc-transitive graphs, these graphs have remained an active topic of research to this day; see [7,8,11,12,14,15,16,18,20,21,22,23,24,25,26,27,33,35].…”

“…The intersection of two adjacent G-alternating cycles is called a G-attachment set. The attachment of alternating cycles concept has been addressed in a number of papers [18,25,26,27,33] with a particular attention given to the so called G-tightly attached graphs, that is, graphs where two adjacent G-alternating cycles have every other vertex in common. In other words, their Gattachment number coincides with G-radius.…”

“…In all the above definitions the symbol G is omitted when G = AutX. Tightly attached graphs with odd radius have been completely classified in [18], whereas the classification of tightly attached graphs with even radius, dealt with also in [25,33], has been very recently completed in [27]. At the other extreme, graphs with G-attachment number equal to 1 and 2, respectively, are called G-loosely attached graphs and G-antipodally attached graphs.…”

Smallest tetravalent half-arc-transitive graphs with the vertex-stabiliser isomorphic to the dihedral group of order 8

“…The tetravalent 1 2 -arc-transitive graphs (and half-arc-transitive graphs in general) were much studied by many authors from different points of view, ranging from purely combinatorial [7,13,17,25,26,27,29], geometrical [15], to permutation group theoretical [1,8,10] and abstract group theoretical [16,21,24].…”