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...

We define an on-line (incremental) algorithm that, given a (possibly infinite) pseudo-transitive oriented graph, produces transitive reorientation. This implies that theorem of Ghouila-Houri is provable in RCA 0 and hence computably true.

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...

A family of arc-transitive graphs is studied. The vertices of these graphs are ordered pairs of distinct points from a finite projective line, and adjacency is defined in terms of the cross ratio. A uniform description of the graphs is given, their automorphism groups are determined, the problem of isomorphism between graphs in the family is solved, some combinatorial properties are explored, a...

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.

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...