Independent Sets and Continued Fractions

TL;DR

This paper explores the distribution of the number of independent sets in various graph classes, revealing growth exponents, density results, and a phase transition linked to continued fraction theory.

Contribution

It establishes new bounds on the set of independent set counts in trees and planar graphs, and connects graph enumeration with continued fractions and number theory.

Findings

The set of independent set counts in trees has a growth exponent of 0.1966.

Connected planar graphs can realize almost all positive integers as independent set counts.

A phase transition occurs at a certain edge density, relating to continued fraction properties.

Abstract

Linek's 1989 problem asks whether the numbers of independent sets of trees avoid infinitely many positive integers. We show that the set of natural numbers realized as the number of independent sets of a tree has a lower growth exponent of $0.1966$. We further prove that the set of positive integers representable by connected planar graphs has asymptotic density one. Lastly, we establish a phase transition: the number of independent sets of graphs with fewer than $d|V|$ edges for any $d<1$ is contained in a set of density zero, whereas, following Shkredov's recent breakthrough on Zaremba's conjecture in continued fraction theory, there exists a constant $D$ such that the number of independent sets of graphs with at most $D|V|$ edges covers all positive integers.