Almost all 5‐regular graphs have a 3‐flow

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...

Almost all graphs and h-hypergraphs have small diameter

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 Friendly Matrices Have Many Obstructions

A symmetric m×m matrix M with entries taken from {0, 1, ∗} gives rise to a graph partition problem, asking whether a graph can be partitioned into m vertex sets matched to the rows (and corresponding columns) of M such that, if Mij = 1, then any two vertices between the corresponding vertex sets are joined by an edge, and if Mij = 0 then any two vertices between the corresponding vertex sets ar...

Almost All Rooted Maps Have Large Representativity

Let M be a map on a surface S. The edge-width of M is the length of a shortest non-contractible cycle of M . The face-width (or, representativity) of M is the smallest number of intersections a noncontractible curve in S has with M . (The edge-width and face-width of a planar map may be de ned to be in nity.) A map is an LEW-embedding if its maximum face valency is less than its edge-width. For...