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-...

A general method is described which gives rise to highly symmetric tessellations of the Cantor sphere, i.e., the 2-sphere with the Cantor set removed and endowed with the hyperbolic geometry with constant negative curvature. These tessellations correspond to almost vertextransitive planar graphs with infinitely many ends. Their isometry groups have infinitely many ends and are free products wit...

We prove that all connected vertex-transitive graphs of order p, p a prime, can be decomposed into Hamilton cycles.

By definition, Cayley graphs are vertex-transitive, and graphs underlying regular or orientably-regular maps (on surfaces) are arc-transitive. This paper addresses questions about how large the automorphism groups of such graphs can be. In particular, it is shown how to construct 3-valent Cayley graphs that are 5-arc-transitive (in answer to a question by Cai Heng Li), and Cayley graphs of vale...

A regular graph G is called vertex transitive if the automorphism group of G contains a single orbit. In this paper we define and consider another class of regular graphs called neighborhood regular graphs abbreviated NR. In particular, let G be a graph and N [v] be the closed neighborhood of a vertex v of G. Denote by G(N [v]) the subgraph of G induced by N [v]. We call G NR if G(N [v]) ∼= G(N...

We classify trivalent vertex-transitive graphs whose edge sets have a partition into 2-factor composed of two cycles and 1-factor that is invariant under the action automorphism group.