On the number of maximal independent sets: From Moon-Moser to Hujter-Tuza

TL;DR

This paper extends classical extremal graph theory results by determining the maximum number of maximal independent sets in graphs excluding certain induced subgraphs, unifying and generalizing previous findings.

Contribution

It introduces a new extremal function for graphs with no induced triangle matching of size t+1, generalizing prior results by Moon-Moser and Hujter-Tuza.

Findings

Determined maximum mis_t(n) for graphs without induced triangle matching of size t+1.

Reproved a stability result for triangle-free graphs with no large induced matching.

Unified classical results within a broader extremal graph framework.

Abstract

We connect two classical results in extremal graph theory concerning the number of maximal independent sets. The maximum number mis$(n)$ of maximal independent sets in an $n$-vertex graph was determined by Moon and Moser. The maximum number mis$_\bigtriangleup(n)$ of maximal independent sets in an $n$-vertex triangle-free graph was determined by Hujter and Tuza. We determine the maximum number mis$_t(n)$ of maximal independent sets in an $n$-vertex graph containing no induced triangle matching of size $t+1$. We also reprove a stability result of Kahn and Park on the maximum number mis$_{\bigtriangleup,t}(n)$ of maximal independent sets in an $n$-vertex triangle-free graphs containing no induced matching of size $t+1$.