A graph which is edge transitive but not arc transitive

Abstract

A graph having 2 7 vertices is described, whose automorphism group is transitive on vertices and undirected edges, but not on directed edges.The object of this Note is to provide an example of a finite graphs whose automorphism group is transitive on vertices and undirected edges, but not on directed edges. In other words, the graph is edge transitive but not arc transitive. The question of the existence of such a graph was raised by Tutte on p. 60 of [ 2 ] , and he showed that it must have even valency if it exists. An infinite family of such graphs was constructed by Bouwer in [ 11, one for each valency 2n(n 2 2 ) . His smallest example has 54 vertices and valency 4. The example constructed here has 27 vertices and valency 4, and so it appears to be the smallest known example at present. It has diameter 4 and girth 5 , and it is not bipartite. I have been informed by the referee that a further example of such a graph has been constructed by P. Kornya.We adopt the following procedure. Start with a transitive permutation group on a finite set SZ, denoted by Gn. Let A be an orbit of G on SZ X SZ, other than the diagonal orbit, and let A' be the paired orbit { ( p , a) I (a, p ) E A}. Then G acts transitively on the arcs of the digraph having point set SZ and arc set A. Now let D be the digraph with point set SZ and arc set A U A', and let r be the graph obtained from D by viewing each symmetric pair of arcs {(a,p),(p,a)} as an undirected edge {a$}. If A f A', then G acts transitively on the edges of r, but not on the arcs of D , which are the directed edges of r. The problem is that this does not automatically yield an example of the type that we are seeking, because the full automorphism group of r may be larger than G, and may act transitively on the arcs of D . Indeed, this seems to happen most of the time. The smallest example that I have been able to find in which this does not occur is one in which Gn is a group of order 54 and degree 27, having a regular nonabelian subgroup. The relevant A has

Some recent discoveries about half-arc-transitive graphs

“…The action of G on the blocks of size 3 (which are {1, 55, 56} and its images) gives an epimorphism from G to the group G above, with elementary abelian kernel K of order 3 10 . Let p = a −1 b, q = a −1 pa, r = a −1 qa and s = a −1 ra.…”

“…The smallest possible valency of a half-arc-transitive graph is 4, and this is the valency of the smallest half-arc-transitive graph, a graph on 27 vertices constructed independently by Doyle [7] and Holt [10].…”

On quartic half-arc-transitive metacirculants

“…We note that graphs X o (m, n; r) correspond to the graphs X(r; m, n) introduced in [18]. We also note that the Holt graph, the smallest half-arc-transitive graph (see [1,10,13]), is isomorphic to X o (3, 9; 2). Example 2.2 For each m ≥ 4 even, n ≥ 4 even, r ∈ Z * n , where r m = 1, and t ∈ Z n , where t(r − 1) = 0, let X e (m, n; r, t) be the graph with vertex set V = {u j i | i ∈ Z m , j ∈ Z n } and edges defined by the following adjacencies:…”

“…The smallest graph in Bouwer's family has 54 vertices and valency 4. Doyle [10] and Holt [13] independently found one with 27 vertices, a graph that is now known to be the smallest half-arc-transitive graph [1].…”

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“…The smallest graph in his family has 54 vertices. In 1981 Holt [10] found an example with 27 vertices and degree 4. Recently, Alspach et al [2] proved that Holt's graph is the smallest 1/2-transitive graph in the sense that there are no 1/2-transitive graphs with fewer than 27 vertices or with degree less than 4.…”