Almost all quasigroups have rank 2

Almost all webs are not rank-perfect

Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [8,12] and claw-free graphs [7,8]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [9]. Ben Rebea conjectured a description for quasi-line graphs, see [12]; Chudnovsky and Seymour [2] v...

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...

Complex Tensors Almost Always Have Best Low-rank Approximations

Low-rank tensor approximations are plagued by a well-known problem — a tensor may fail to have a best rank-r approximation. Over R, it is known that such failures can occur with positive probability, sometimes with certainty: in R2×2×2, every tensor of rank 3 fails to have a best rank-2 approximation. We will show that while such failures still occur over C, they happen with zero probability. I...

Almost all graphs and h-hypergraphs have small diameter