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

Erdös 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 Erdös, 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 ⩾ 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.