An equivalent relation between transitive 1-factorizations of arctransitive graphs and factorizations of their automorphism groups is established. This relation provides a method for constructing and characterizing transitive 1-factorizations for certain classes of arc-transitive graphs. Then a characterization is given of 2-arc-transitive graphs which have transitive 1factorizations. In this c...

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 transitive decomposition is a pair (Γ,P) where Γ is a graph and P is a partition of the arc set of Γ such that there is a subgroup of automorphisms of Γ which leaves P invariant and transitively permutes the parts in P. In an earlier paper we gave a characterisation of G-transitive decompositions where Γ is the graph product Km×Km and G is a rank 3 group of product action type. This character...

A transitive decomposition of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the group preserving the partition is arc-transitive and acts primitively on the parts.

A Cayley (resp. bi-Cayley) graph on a dihedral group is called dihedrant bi-dihedrant). In 2000, classification of trivalent arc-transitive dihedrants was given by Marušič and Pisanski, several years later, non-arc-transitive order 4p or 8p (p prime) were classified Feng et al. As generalization these results, our first result presents dihedrants. Using this, complete vertex-transitive non-Cayl...

Two different constructions are given of a rank 8 arc-transitive graph with 165 vertices and valency 8, whose automorphism group is M11. One involves 3-subsets of an 11-set while the other involves 4-subsets of a 12-set, and the constructions are linked with the Witt designs on 11, 12 and 24 points. Four different constructions are given of a rank 9 arc-transitive graph with 55 vertices and val...

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...

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...

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 taken as vertices. This graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of K4 and as the edge-disjoint union of 21 in...

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 taken as vertices. This graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of K4 and as the edge-disjoint union of 21 i...