The number of maximal independent sets in connected triangle-free graphs

Abstract

Erdos and Moser raised the problem of determining the largest number of maximal independent sets of a general graph G of order n and those graphs achieving this largest number. This problem was solved by Erdos, and later Moon and Moser. It then was extensively studied for various classes of graphs, including trees, forests, (connected) graphs with at most one cycle, bipartite graphs, connected graphs, k-connected graphs and triangle-free graphs. This paper studies the problem for connected triangle-free graphs. In particular, we prove that every connected triangle-free graph of order n greater than or equal to 22 has at most 5 . 2((n-6)/2) (respectively, 2((n-1)/2)) maximal independent sets if n is even (respectively, odd). Extremal graphs achieving this maximum value are also characterized. (C) 1999 Elsevier Science B.V. All rights reserved.