Let0 be a graph and G be a 2-arc transitive automorphism group of0. For a vertex x ∈ 0 let G(x)0(x) denote the permutation group induced by the stabilizer G(x) of x in G on the set 0(x) of vertices adjacent to x in 0. Then 0 is said to be a locally projective graph of type (n, q) if G(x)0(x) contains PSLn(q) as a normal subgroup in its natural doubly transitive action. Suppose that 0 is a local...

IS rel)re:se!otlng vertex-transitive as coset to allow us to obtain new results eral older results concerning the numbers transitive that are not 1 studied for more than a now, vertex-transitive graphs have quite great deal of activity. Their rich groups of automorphisms make from both the point of view of groups as well as of combinatorics. Among the most recent achievements, we can mention th...

A circulant is a Cayley digraph over a finite cyclic group. The classification of arc-transitive circulants is shown. The result follows from earlier descriptions of Schur rings over cyclic groups.

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 (hereditarily finite) set/hyperset S can be completely depicted by a (finite pointed) graph GS—dubbed its membership graph— in which every node represents an element of the transitive closure of {S} and every arc represents a membership relation holding between its source and its target. In a membership graph different nodes must have different sets of successors and, more generally, if the g...

A new infinite family of bipartite cubic 3-arc transitive graphs is constructed and studied. They provide the first known examples admitting a 2-arc transitive vertex-biquasiprimitive group of automorphisms for which the stabiliser of the biparts is not quasiprimitive on either bipart.

We establish natural bijections between three different classes of combinatorial objects; namely certain families of locally 2–arc transitive graphs, partial linear spaces, and homogeneous factorisations of arc-transitive graphs. Moreover, the bijections intertwine the actions of the relevant automorphism groups. Thus constructions in any of these areas provide examples for the others.

In this paper, we determine all connected 5-arc transitive cubic Cayley graphs on the alternating group A47; there are only two such graphs (up to isomorphism). By earlier work of the authors, these are the only two nonnormal connected cubic arc-transitive Cayley graphs for finite nonbelian simple groups, and so this paper completes the classification of such non-normal Cayley graphs.

A digraph is semicomplete if any two vertices are connected by at least one arc and locally the out-neighbourhood in-neighbourhood of vertex induce a digraph. In this paper we study various subclasses digraphs for which give structural decomposition theorems. As consequence obtain several applications, among an answer to conjecture Naserasr first third authors (Aboulker et al., 2021): oriented ...

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...