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Constraints on the number of maximal independent sets in graphs

✍ Scribed by Jiuqiang Liu

Publisher: John Wiley and Sons
Year: 1994
Tongue: English
Weight: 387 KB
Volume: 18
Category: Article
ISSN: 0364-9024
DOI: 10.1002/jgt.3190180211

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✦ Synopsis

Abstract

A maximal independent set of a graph G is an independent set that is not contained properly in any other independent set of G. Let i(G) denote the number of maximal independent sets of G. Here, we prove two conjectures, suggested by P. Erdös, that the maximum number of maximal independent sets among all graphs of order n in a family Φ is o(3^n/3^) if Φ is either a family of connected graphs such that the largest value of maximum degrees among all graphs of order n in Φ is o(n) or a family of graphs such that the approaches infinity as n → ∞.

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