Let Γ be a finite digraph and let G be a subgroup of the automorphism group of Γ. A directed cycle ~ C of Γ is called G-consistent whenever there is an element of G whose restriction to ~ C is the 1-step rotation of ~ C. Consistent cycles in finite arc-transitive graphs were introduced by Conway in one of his public lectures. He observed that the number of G-orbits of G-consistent cycles of an ...

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...

It is shown that every connected vertex-transitive graph of order 4p, where p is a prime, is hamiltonian with the exception of the Coxeter graph which is known to possess a Hamilton path. In 1969, Lovász [22] asked if every finite, connected vertex-transitive graph has a Hamilton path, that is, a path going through all vertices of the graph. With the exception of K 2 , only four connected verte...

A (di)graph is supereulerian if it contains a spanning, connected, eulerian sub(di)graph. This property is a relaxation of hamiltonicity. Inspired by this analogy with hamiltonian cycles and by similar results in supereulerian graph theory, we give a number of sufficient Ore type conditions for a digraph to be supereulerian. Furthermore, we study the following conjecture due to Thomassé and the...

This paper develops methods to reason about graph transformations, and in particular to show that transitivity and reachability invariants are preserved during transformations. In our approach, graph transformations consist of a pattern defining an applicability condition, and an operational description of the desired transformation. Whereas previous work was restricted to Boolean combinations ...

The notion of a C-ultrahomogeneous (or C-UH) graph due to D. Isaksen et al. is adapted for digraphs and applied to the cubic distance-transitive graphs considered both as graphs and digraphs when C is formed by shortest cycles and (k − 1)-paths with k = arc-transitivity. Moreover, (k − 1)powers of shortest cycles taken with orientation assignments that make these graphs become C-UH digraphs are...

Let A be the adjacency matrix of a graph Γ . The nullity of A (that is, the dimension of the nullspace of A), when viewed as a matrix over a field of prime characteristic p, is called the p-nullity of Γ . We present several families of arc-transitive graphs with arbitrarily large p-nullity. We also show that the p-nullity of a vertextransitive graph of order a power of p is zero, provided that ...