A homogeneous factorisation of a digraph Γ consists of a partition P = {P1, . . . , Pk} of the arc set AΓ and two vertex-transitive subgroups M 6 G 6 Aut(Γ) such that M fixes each Pi setwise while G leaves P invariant and permutes its parts transitively. Given two graphs Γ1 and Γ2 we consider several ways of taking a product of Γ1 and Γ2 to form a larger graph, namely the direct product, cartes...

A connguration is weakly ag-transitive if its group of automor-phisms acts intransitively on ags but the group of all automorphisms and anti-automorphisms acts transitively on ags. It is shown that weakly ag-transitive conngurations are in one-to-one correspondence with bipartite 1 2-arc-transitive graphs of girth not less than 6. Several innnite families of weakly ag-transitive conngurations a...

The notion of a C-ultrahomogeneous graph, due to Isaksen et al., is adapted for digraphs and studied for the twelve cubic distance transitive graphs, with C formed by g-cycles and (k − 1)-paths, where g = girth and k = arc-transitivity. Excluding the Petersen, Heawood and Foster (90 vertices) graphs, one can go further by considering the (k− 1)-powers of g-cycles under orientation assignments p...

The transitive closure (or reachability) problem in a directed graph consists in finding whether there is a path between any two vertices. In this paper, we first study the problem of parallelization of Italiano's algorithm for dynamic updating the transitive closure after inserting a new arc into the graph represented as a list of arcs. To this end, by means of the data structure first propose...

are considered in this paper. Examples of finite Qn-geometries are constructed for any n i> 2. In particular, we obtain new examples of finite 2-arc-transitive graphs of girth 5, containing Petersen subgraphs (cf. [4], [5] for an enquiry into this class of graphs). Furthermore, we give a complete classification of Q2-geometries using coset enumeration. In particular, we show that they are all f...

Every finite, self-dual, regular (or chiral) 4-polytope of type {3, q, 3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edgebut not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope.

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...