Almost all Cayley graphs have diameter 2

Almost all graphs and h-hypergraphs have small diameter

Diameter 2 Cayley graphs of dihedral groups

We consider the degree-diameter problem for Cayley graphs of dihedral groups. We find upper and lower bounds on the maximum number of vertices of such a graph with diameter 2 and degree d. We completely determine the asymptotic behaviour of this class of graphs by showing that both limits are asymptotically d/2.

Almost Diameter Rigidity for Cayley Plane

In this paper we give a generalisation of the Radius Rigidity theorem of F.Wilhelm. This is done by showing that if a Riemannian submersion of S15 with 7-dimensional fibres has at least one fibre which is a great sphere then all the fibres are so. Some weaker than known conditions which force the existence of such a fibre are also discussed.

Almost all H-free graphs have the Erdős-Hajnal property

Erdős and Hajnal conjectured that, for every graph H, there exists a constant (H) > 0 such that every H-free graph G (that is, not containing H as an induced subgraph) must contain a clique or an independent set of size at least |G| . We prove that there exists (H) such that almost every H-free graph G has this property, meaning that, amongst the H-free graphs with n vertices, the proportion ha...

Almost All Cop-win Graphs Have a Universal Vertex

Cops and Robbers is a vertex-pursuit game played on graphs which has received much recent attention. Graphs where one cop has a winning strategy, so-called cop-win graphs, were discovered in the early 1980’s by Nowakowski and Winkler and independently by Quilliot. Since their introduction, the structure of cop-win graphs has been relatively well-understood. Cop-win graphs possess a vertex elimi...