A decomposition of a graph is a partition of the edge set. One can also look at partitions of the arc set but in this talk we restrict our attention to edges. If each part of the decomposition is a spanning subgraph then we call the decomposition a factorisation and the parts are called factors. Decompositions are especially interesting when the subgraphs induced by each part are pairwise isomo...

A near-polygonal graph is a graph Γ which has a set C of m-cycles for some positive integer m such that each 2-path of Γ is contained in exactly one cycle in C. If m is the girth of Γ then the graph is called polygonal. We introduce a method for constructing near-polygonal graphs with 2-arc transitive automorphism groups. As special cases, we obtain several new infinite families of polygonal gr...

A graph is symmetric or arc-transitive if its automorphism group acts transitively on vertices, edges and arcs. Let p, q be odd primes with p, q ≥ 5 and X a q-valent symmetric graph of order 4p. In this paper, we proved that X K4p with 4p-1=q, X K2p,2p-2pK2 with 2p-1=q, the quotient graph of X is isomorphic to Kp,p and p=q, or K2p and 2p-1=q.

A construction is given for an infinite family {0n} of finite vertex-transitive non-Cayley graphs of fixed valency with the property that the order of the vertex-stabilizer in the smallest vertex-transitive group of automorphisms of 0n is a strictly increasing function of n. For each n the graph is 4-valent and arc-transitive, with automorphism group a symmetric group of large prime degree p> 2...

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Letn be a product of three primes. The problemon the classification of the tetravalent half-arc-transitive graphs of order n has been considered by Xu (1992), Feng et al. (2007) and Wang and Feng (2010), and it was solved for the cases where n is a prime cube or twice a produ...

Middendorf, M. and F. Pfeiffer, Weakly transitive orientations, Hasse diagrams and string graphs, Discrete Mathematics 111 (1993) 393-400. We introduce the notion of a weakly transitive orientation for graphs as a natural generalization of transitive orientations and give a characterization for weakly transitive orientations in terms of forbidden substructurs. As a corollary of this characteriz...

Transitive permutation groups having a non self paired suborbit of length are investigated via the corresponding orbital graphs If G is such a group and X is the orbital graph associated with a sub orbit of length of G which is not self paired then X has valency and admits a vertex and edge but not arc transitive action of G There is a natural balanced orientation of the edge set of X induced a...

The present paper is devoted classification of $\mathrm {AT4}(p,p+2,r)$-graphs. There a unique {AT4}(p,p+2,r)$-graph with $p=2$, namely, the distance-transitive Soicher graph intersection array $\{56, 45, 16, 1;1, 8, 56\}$, whose local graphs are isomorphic to Gewirtz graph. existence an ${p>2}$ remains open question. It known that each strongly regular parameters $\big ((p+2)(p^2+4p+2),p(p+3),...

Two different constructions are given of a rank 8 arc-transitive graphwith 165 vertices and valency 8,whose automorphismgroup isM11. One involves 3-subsets of an 11-set while the other involves 4-subsets of a 12-set, and the constructions are linkedwith theWitt designs on 11, 12 and 24 points. Four different constructions are given of a rank 9 arc-transitive graph with 55 vertices and valency 6...