Lower bounds on the maximum number of non-crossing acyclic graphs

TL;DR

This paper establishes new exponential lower bounds on the maximum number of non-crossing spanning trees and forests on point sets, especially the double chain configuration, using analytic combinatorics.

Contribution

It provides improved lower bounds for the maximum counts of non-crossing spanning trees and forests, and introduces new upper bounds for the double chain configuration.

Findings

Omega(12.52^N) non-crossing spanning trees on double chain

Omega(13.61^N) non-crossing forests on double chain

O(22.12^N) upper bound for non-crossing spanning trees on double chain

Abstract

This paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of non-crossing spanning trees and forests. We show that the so-called double chain point configuration of N points has Omega(12.52^N) non-crossing spanning trees and Omega(13.61^N) non-crossing forests. This improves the previous lower bounds on the maximum number of non-crossing spanning trees and of non-crossing forests among all sets of N points in general position given by Dumitrescu, Schulz, Sheffer and T\'oth. Our analysis relies on the tools of analytic combinatorics, which enable us to count certain families of forests on points in convex position, and to estimate their average number of components. A new upper bound of O(22.12^N) for the number of non-crossing spanning trees of the double chain is also obtained.