Almost all digraphs have a kernel

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...

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 Integer Matrices Have No Integer Eigenvalues

In a recent issue of this MONTHLY, Hetzel, Liew, and Morrison [4] pose a rather natural question: what is the probability that a random n× n integer matrix is diagonalizable over the field of rational numbers? Since there is no uniform probability distribution on Z, we need to exercise some care in interpreting this question. Specifically, for an integer k ≥ 1, let Ik = {−k,−k + 1, . . . , k − ...