Minimal matchings of point processes

Trees and Matchings from Point Processes

A factor graph of a point process is a graph whose vertices are the points of the process, and which is constructed from the process in a deterministic isometry-invariant way. We prove that the d-dimensional Poisson process has a one-ended tree as a factor graph. This implies that the Poisson points can be given an ordering isomorphic to the usual ordering of the integers in a deterministic iso...

Ramsey-minimal saturation numbers for matchings

Given a family of graphs F , a graph G is F-saturated if no element of F is a subgraph of G, but for any edge e in G, some element of F is a subgraph of G + e. Let sat(n,F) denote the minimum number of edges in an F-saturated graph of order n, which we refer to as the saturation number or saturation function of F . If F = {F}, then we instead say that G is F -saturated and write sat(n, F ). For...

On the Intersection Number of Matchings and Minimum Weight Perfect Matchings of Multicolored Point Sets

Let P and Q be disjoint point sets with 2k and 2l elements respectively, and M1 and M2 be their minimum weight perfect matchings (with respect to edge lengths). We prove that the edges of M1 and M2 intersect at most |M1| + |M2| − 1 times. This bound is tight. We also prove that P and Q have perfect matchings (not necessarily of minimum weight) such that their edges intersect at most min{r, s} t...

Point sets of minimal energy

Realizability of point processes

There are various situations in which it is natural to ask whether a given collection of k functions, ρj(r1, . . . , rj), j = 1, . . . , k, defined on a set X, are the first k correlation functions of a point process on X. Here we describe some necessary and sufficient conditions on the ρj ’s for this to be true. Our primary examples are X = Rd, X = Zd, and X an arbitrary finite set. In particu...