### Near-perfect non-crossing harmonic matchings in randomly labeled points on a circle

*József Balogh, Boris Pittel, Gelasio Salazar*

#### Abstract

Consider a set S of points in the plane in convex position, where each point has an integer label from {0,1,…,n-1}. This naturally induces a labeling of the edges: each edge (i,j) is assigned label i+j, modulo n. We propose the algorithms for finding large non--crossing

*harmonic*matchings or paths, i. e. the matchings or paths in which no two edges have the same label. When the point labels are chosen uniformly at random, and independently of each other, our matching algorithm with high probability (w.h.p.) delivers a nearly--perfect matching, a matching of size n/2 - O(n1/3*ln*n).Full Text: GZIP Compressed PostScript PostScript PDF original HTML abstract page