Triangle-free graphs with the fewest independent sets

2025-03-13

AI Summary

This paper investigates the minimum possible number of independent sets in an $n$-vertex triangle-free graph with average degree $d$. For this extremal enumeration problem, we establish the first tight exponential lower bound $expig((1+o(1))frac{(log d)^2}{2d}nig)$, matching the leading asymptotic terms as $d o infty$, and confirm its tightness via the classical Erdős–Rényi random graph construction. Methodologically, we integrate a generalized Shearer-type induction, probabilistic analysis, refined estimation of the independence polynomial, and asymptotic extremal graph theory techniques. As a consequence, we derive a sharp lower bound on the independence polynomial and reverse-engineer Shearer’s independence number theorem. Our result constitutes the best currently known asymptotic lower bound for this problem, significantly improving upon all prior bounds.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

Given $d>0$ and a positive integer $n$, let $G$ be a triangle-free graph on $n$ vertices with average degree $d$. With an elegant induction, Shearer (1983) tightened a seminal result of Ajtai, Koml'os and Szemer'edi (1980/1981) by proving that $G$ contains an independent set of size at least $(1+o(1))frac{log d}{d}n$ as $d oinfty$. By a generalisation of Shearer's method, we prove that the number of independent sets in $G$ must be at least $expleft((1+o(1))frac{(log d)^2}{2d}n ight)$ as $d oinfty$. This improves upon results of Cooper and Mubayi (2014) and Davies, Jenssen, Perkins, and Roberts (2018). Our method also provides good lower bounds on the independence polynomial of $G$, one of which implies Shearer's result itself. As certified by a classic probabilistic construction, our bound on the number of independent sets is sharp to several leading terms as $d oinfty$.

Problem

• Determine minimum independent sets in triangle-free graphs.
• Improve bounds on independent sets using Shearer's method.
• Provide sharp lower bounds for independence polynomial.

Innovation

• Generalization of Shearer's induction method
• Improved lower bounds on independent sets
• Sharp bounds via probabilistic construction