Non-Crossing Perfect Matchings and Triangle-Free Geometric Graphs

TL;DR

This paper investigates the maximum number of edge-disjoint non-crossing perfect matchings on planar point sets that form triangle-free geometric graphs, providing conditions and bounds for their existence.

Contribution

It introduces new conditions and bounds for the existence of multiple edge-disjoint non-crossing perfect matchings forming triangle-free graphs in various point configurations.

Findings

Established a sufficient condition for n such matchings in general position.

Derived a lower bound on the number of such matchings.

Analyzed different point set configurations for these matchings.

Abstract

We study extremal type problem arising from the question: What is the maximum number of edge-disjoint non-crossing perfect matchings on a set S of 2n points in the plane such that their union is a triangle-free geometric graph? We approach this problem by considering four different situations of S. In particular, in the general position, we obtain (i) a sufficient condition for the existence of n edge-disjoint non-crossing perfect matchings in the general position whose union is a maximal triangle-free geometric graph, and (ii) a lower bound on the number of edge-disjoint non-crossing perfect matchings whose union is a triangle free geometric graph.