In this paper, a characterisation is given of finite s-arc transitive Cayley graphs with s ≥ 2. In particular, it is shown that, for any given integer k with k ≥ 3 and k 6= 7, there exists a finite set (maybe empty) of s-transitive Cayley graphs with s ∈ {3, 4, 5, 7} such that all s-transitive Cayley graphs of valency k are their normal covers. This indicates that s-arc transitive Cayley graphs...

A basic result in Ramsey theory states that any tournament contains a “large” transitive subgraph. Since transitive tournaments contain only transitive subgraphs, it is natural to ask which subgraphs must appear in any large tournament that is “far” from being transitive. One result of this type was obtained by Fox and Sudakov who characterized the tournaments that appear in any tournament that...

A substitution group G of degree n is said to be r-fold transitive if each of the n(n — 1) • • • (n—r + 1) permutations of its n letters taken r at a time is represented by at least one substitution of G. It is not sufficient to say that each of its possible sets of r letters is transformed into every one of these sets by the substitutions of G. For instance, in each of the semi-metacyclic grou...

We classify the neighbour-transitive codes in Johnson graphs J(v, k) of minimum distance at least 3 which admit a neighbour-transitive group of automorphisms that is an almost simple 2-transitive group of degree v and does not occur in an infinite family of 2-transitive groups. The result of this classification is a table of 22 codes with these properties. Many have relatively large minimum dis...

This paper presents a classification of vertex-primitive and vertex-biprimitive 2-path-transitive graphs which are not 2-arc-transitive. The classification leads to constructions of new examples of half-arc-transitive graphs.

For a fixed k, an oriented graph is k-transitive if it is acyclic and for every directed path p=u1→u2→...→uk+2 with k+2 vertices, p induces a clique if each of the two subpaths u1→u2→...→uk+1 and u2→...→uk+2 induces a clique. We describe an algorithm to find a maximum weight clique in a k-transitive graph. Consider a hereditary family G of graphs. A graph H(V,E) is called G-k-mixed if its edge ...

we find recursive formulae for the number of perfect matchings in a graph g by splitting g into subgraphs h and q. we use these formulas to count perfect matching of p hypercube qn. we also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in g, is the graph constructed from by deleting edges with an en...

The existence of a connected 12-regular {K4,K2,2,2}-ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane as vertices and an adjacency resembling that of star Cayley graphs. The graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copi...