A cycle decomposition of a graph Γ is a set C of cycles of Γ such that every edge of Γ belongs to exactly one cycle in C. Such a decomposition is called arc-transitive if the group of automorphisms of Γ that preserve C setwise acts transitively on the arcs of Γ . In this paper, we study arc-transitive cycle decompositions of tetravalent graphs. In particular, we are interested in determining an...

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple (v, u, x, y) of vertices such that both (v, u, x) and (u, x, y) are paths of length two. The 3-arc graph of a graph G is defined to have vertices the arcs of G such that two arcs uv, xy are adjacent if and only if (v, u, x, y) is a 3-arc of G. In this paper we prove that any connected 3-arc graph is Hamiltonian, and all iterative 3-...

Let Γ be a graph admitting an arc-transitive subgroup G of automorphisms that leaves invariant a vertex partition B with parts of size v ≥ 3. In this paper we study such graphs where: for B, C ∈ B connected by some edge of Γ , exactly two vertices of B lie on no edge with a vertex of C; and as C runs over all parts of B connected to B these vertex pairs (ignoring multiplicities) form a cycle. W...

A graph G is balanced if the maximum ratio of edges to vertices, taken over all subgraphs of G, occurs at G itself. This note uses the maxow/min-cut theorem to prove a good characterization of balanced graphs. This characterization is then applied to some results on how balanced graphs may be combined to form a larger balanced graph. In particular, we show that edge-transitive graphs and comple...

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...

A regular edge but not vertex transitive graph is said to be semisym metric The study of semisymmetric graphs was initiated by Folkman who among others gave constructions of several in nite families such graphs In this paper a generalization of his construction for or ders a multiple of is proposed giving rise to some new families of semisymmetric graphs In particular one associated with the cy...

Resolving parameters are a fundamental area of combinatorics with applications not only to many branches but also other sciences. In this study, we construct class Toeplitz graphs and will be denoted by T 2 n W ...

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...

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...