English

Improving Sharir and Welzl's bound on crossing-free matchings through solving a stronger recurrence

Combinatorics 2017-01-25 v2

Abstract

Sharir and Welzl [1] derived a bound on crossing-free matchings primarily based on solving a recurrence based on the size of the matchings. We show that the recurrence given in Lemma 2.3 in Sharir and Welzl can be improve to (2n6s)Mam(P)683(s+2)Mam1(P)(2n-6s)\textbf{Ma}_{m}(P)\leq\frac{68}{3}(s+2)\textbf{Ma}_{m-1}(P) and (3n7s)Mam(P)44.5(s+2)Mam1(P)(3n-7s)\textbf{Ma}_{m}(P)\leq44.5(s+2)\textbf{Ma}_{m-1}(P), thereby improving the upper bound for crossing-free matchings.

Keywords

Cite

@article{arxiv.1701.05909,
  title  = {Improving Sharir and Welzl's bound on crossing-free matchings through solving a stronger recurrence},
  author = {Chenchao You},
  journal= {arXiv preprint arXiv:1701.05909},
  year   = {2017}
}