Half-Transitive Graphs of Prime-Cube Order
Journal of Algebraic Combinatorics: An International Journal
1/2-Transitive Graphs of Order 3p
Journal of Algebraic Combinatorics: An International Journal
A 1/2-transitive graph of valency 4 with a nonsolvable group of automorphisms
Journal of Graph Theory
The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
Automorphism groups and isomorphisms of Cayley digraphs
Discrete Mathematics - Special issue on Graph theory
Maps and half-transitive graphs of valency 4
European Journal of Combinatorics
Half-transitive group actions on finite graphs of valency 4
Journal of Combinatorial Theory Series B
Constructing 4-valent 12 -transitive graphs with a nonsolvable automorphism group
Journal of Combinatorial Theory Series B
Tetravalent graphs admitting half-transitive group actions: alternating cycles
Journal of Combinatorial Theory Series B
Weekly flag-transitive configurations and half-arc-transitive graphs
European Journal of Combinatorics
On half-transitive metacirculant graphs of prime-power order
Journal of Combinatorial Theory Series B
Constructing ½-arc-transitive graphs of valency 4 and vertex stabilizer Z2× Z2
Discrete Mathematics
Bridging semisymmetric and half-arc-transitive actions on graphs
European Journal of Combinatorics
A tetravalent half-arc-transitive graph with non-abelian vertex stabilizer
Journal of Combinatorial Theory Series B
Classifying Arc-Transitive Circulants
Journal of Algebraic Combinatorics: An International Journal
Tetravalent half-transitive graphs of order 4p
European Journal of Combinatorics
Primitive half-transitive graphs constructed from the symmetric groups of prime degrees
Journal of Combinatorial Theory Series B
Reachability relations in digraphs
European Journal of Combinatorics
Journal of Graph Theory
A classification of tightly attached half-arc-transitive graphs of valency 4
Journal of Combinatorial Theory Series B
On quartic half-arc-transitive metacirculants
Journal of Algebraic Combinatorics: An International Journal
Hexavalent half-arc-transitive graphs of order 4p
European Journal of Combinatorics
Half-Edge-Transitive Graphs and Non-Normal Cayley Graphs
Journal of Graph Theory
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The action of a subgroup G of automorphisms of a graph X is said to be half-arc-transitive if it is vertex-transitive and edge-transitive but not arc-transitive. In this case the graph X is said to be G-half-arc-transitive. In particular, when G=AutX the graph X is said to be half-arc-transitive. Two oppositely oriented digraphs may be associated with any such graph in a natural way. The reachability relation of a graph admitting a half-arc-transitive group action is an equivalence relation defined on either of these two digraphs as follows. An arc e is reachable from an arc e^' if there exists an alternating path starting with e and ending with e^'. The reachability relation is clearly universal for all vertex-primitive half-arc-transitive graphs. The smallest known vertex-primitive half-arc-transitive graphs have valency 14 and no such graphs of valency smaller than 10 exist. The natural framework for the question of the existence of half-arc-transitive graphs with universal reachability relation is therefore the family of vertex-imprimitive half-arc-transitive graphs, and in particular those of valency less than 14. It is known that no such graph of valency 4 exists (see D. Marusic, Half-transitive group actions on finite graphs of valency 4, J. Combin. Theory Ser. B 73 (1998) 41-76). In this paper an infinite family of vertex-imprimitive half-arc-transitive graphs of valency 12 with universal reachability relation is constructed. These graphs have a solvable automorphism group but are not metacirculants.