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...

for two normal edge-transitive cayley graphs on groups h and k which have no common direct factor and gcd(jh=h ′j; jz(k)j) = 1 = gcd(jk=k ′j; jz(h)j), we consider four standard products of them and it is proved that only tensor product of factors can be normal edge-transitive.

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...

In this paper, locally 3-arc-transitive regular covers of complete bipartite graphs are studied, and results are obtained that apply to arbitrary covering transformation groups. In particular, methods are obtained for classifying the locally 3-arctransitive graphs with a prescribed covering transformation group, and these results are applied to classify the locally 3-arc-transitive regular cove...

In this note, we construct bipartite 2-walk-regular graphs with exactly 6 distinct eigenvalues as the point-block incidence graphs of group divisible designs with the dual property. For many of them, we show that they are 2-arc-transitive dihedrants. We note that some of these graphs are not described in Du et al. (2008), in which they classified the connected 2-arc transitive dihedrants.

We show that a complete arc K in the projective plane PG(2, q) admitting a transitive primitive group of projective transformations is either a cyclic arc of prime order or a known arc. If the completeness assumption is dropped, then K has either an affine primitive group, or K is contained in an explicit list. In order to find these primitive arcs, it is necessary to determine all complete k-a...

In this paper, we classify the regular embeddings of arc-transitive simple graphs of order pq for any two primes p and q (not necessarily distinct) into orientable surfaces. Our classification is obtained by direct analysis of the structure of arc-regular subgroups (with cyclic vertex-stabilizers) of the automorphism groups of such graphs. This work is independent of the classification of primi...

A graph X is said to be 1 2-transitive if its automorphism group Aut X acts vertex-and edge-but not arc-transitively on X. Then Aut X induces an orientation of the edges of X. If X has valency 4, then this orientation gives rise to so called alternating cycles, that is even length cycles in X whose every other vertex is the head and every other vertex is the tail of its two incident edges in th...

We initiate a general study of what we call orientation completion problems. For a fixed class C of oriented graphs, the orientation completion problem asks whether a given partially oriented graph P can be completed to an oriented graph in C by orienting the (non-oriented) edges in P . Orientation completion problems commonly generalize several existing problems including recognition of certai...