In this paper, we give a characterisation for class of edge-transitive Cayley graphs and provide method constructing valency 4 with arbitrarily large vertex stabiliser. particular, in the last section, obtain certain extensions results Li et al. (Tetravalent odd number vertices, J Comb Theory Ser B 96:164–181, 2006) on half-transitive graphs.

Let D be a locally finite, connected, 1-arc transitive digraph. It is shown that the reachability relation is not universal in D provided that the stabilizer of an edge satisfies certain conditions which seem to be typical for highly arc transitive digraphs. As an implication, the reachability relation cannot be universal in highly arc transitive digraphs with prime inor out-degree. Two differe...

The Goldschmidt-Sims conjecture asserts that there is a nite number of (conjugacy classes of) edge transitive lattices in the automorphism group of a regular tree with prime valence. We prove a similar theorem for irreducible lattices, transitive on the 2-cells of the product of two regular trees of prime valences.

If the relation E is transitive, i.e., if (x, y) E E and (y, z) E E imply that (x, z) E E, we speak of a transitive tournament or a total order. Clearly (Z, <) is a total order. We tend to visualize tournaments by considering every edge (x, y) E E as an arrow leading from x to y. In this sense every tournament can also be considered as a complete graph in which every edge is oriented in some di...

The complementary prism of a graph Γ is the ΓΓ̄, which formed from union and its complement Γ̄ by adding an edge between each pair identical vertices in Γ̄. Vertex-transitive self-complementary graphs provide vertex-transitive prisms. It was recently proved author that ΓΓ̄ core, i.e. all endomorphisms are automorphisms, whenever vertex-transitive, self-complementary, either core or complete graph. ...

We study homogeneous, independent percolation on general quasi-transitive graphs. We prove that in the disorder regime where all clusters are finite almost surely, in fact the expectation of the cluster size is finite. This extends a well-known theorem by Menshikov and Aizenman & Barsky to all quasi-transitive graphs. Moreover we deduce that in this disorder regime the cluster size distribution...

It is known that every closed orientable 3-manifold can be represented by coloured knots ((4], 5]), edge-coloured graphs ((2]) or transitive permutation pairs ((6]). The present paper describes some relations between these representation theories: in particular, it is shown how to obtain a transitive permutation pair representing a 3-manifold M 3 , starting from a coloured knot representing M 3 .

We prove that a tournament with n vertices has more than 0.13n(1 + o(1)) edge-disjoint transitive triples. We also prove some results on the existence of large packings of k-vertex transitive tournaments in an n-vertex tournament. Our proofs combine probabilistic arguments and some powerful packing results due to Wilson and to Frankl and Rödl.