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

Abstract

AbstractWe connect two classical results in extremal graph theory concerning the number of maximal independent sets. The maximum number of maximal independent sets in an ‐vertex graph was determined by Miller and Muller and independently by Moon and Moser. The maximum number of maximal independent sets in an ‐vertex triangle‐free graph was determined by Hujter and Tuza. We give a common generalization of these results by determining the maximum number of maximal independent sets in an ‐vertex graph containing no induced triangle matching of size . This also improves a stability result of Kahn and Park on . Our second result is a new (short) proof of a second stability result of Kahn and Park on the maximum number of maximal independent sets in ‐vertex triangle‐free graphs containing no induced matching of size .