We present an order-theoretic approach to the study of countably infinite locally 2-arc-transitive bipartite graphs. Our approach is motivated by techniques developed by Warren and others during the study of cycle-free partial orders. We give several new families of previously unknown countably infinite locally-2-arc-transitive graphs, each family containing continuum many members. These exampl...

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 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...

In this paper, we characterise the family of finite arc-transitive bicirculants. We show that every bicirculant is a normal r $r$ -cover an graph lies in one eight infinite families or seven sporadic graphs. Moreover, each these ‘basic’ graphs either circulant, and latter case has for some integer .

A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and s-regular if its automorphism group acts regularly on the set of s-arcs of Γ. Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is s-regular for some s ≤ 5. Djokovič and Miller (1980) proved that there are seven types of arc-transitive group action on finite cubic graphs, characterised...

A generalization of some of Folkman’s constructions (see (1967) J. Comb. Theory, 3, 215–232) of the so-called semisymmetric graphs, that is regular graphs which are edgebut not vertex-transitive, was given by Marušič and Potočnik (2001, Europ. J. Combinatorics, 22, 333–349) together with a natural connection between graphs admitting 1 2 -arc-transitive group actions and certain graphs admitting...

Abstract We consider arc colourings of oriented graphs such that for each vertex the colours all out-arcs incident with and in-arcs form intervals. prove existence a colouring is an NP-complete problem. give solution problem r -regular graphs, transitive tournaments, small maximum degree, order some other classes graphs. state conjecture graph there exists consecutive colourable orientation con...