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

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

Published:2023-04-23 Issue:2 Volume:104 Page:440-445
ISSN:0364-9024
Container-title:Journal of Graph Theory
language:en
Short-container-title:Journal of Graph Theory

Author:

Palmer Cory¹^ORCID,Patkós Balázs²^ORCID

Affiliation:

1. Department of Mathematical Sciences University of Montana Missoula Montana USA

2. Department of Combinatorics and its Applications Alfréd Rényi Institute of Mathematics Budapest Hungary

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 .

Funder

Simons Foundation

Nemzeti Kutatási Fejlesztési és Innovációs Hivatal

Publisher

Wiley

Subject

Geometry and Topology,Discrete Mathematics and Combinatorics

Link

https://onlinelibrary.wiley.com/doi/pdf/10.1002/jgt.22971

Reference14 articles.

1. Sharp bound on the number of maximal sum-free subsets of integers

2. The number of maximal sum-free subsets of integers

3. Independent sets in hypergraphs

4. The number of the maximal triangle-free graphs

5. The number of maximal independent sets in connected graphs