A note on the independent domination number of subset graph

The independent domination number i(G) (independent number (G)) is the minimum (maximum) cardinality among all maximal independent sets of G. Haviland (1995) conjectured that any connected regular graph G of order n and degree 1/2n satisfies i(G) 2n/3 1/2. For 1 k l m, the subset graph S _m (k, l) is the bipartite graph whose vertices are the k- and l-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. In this paper, we give a sharp upper bound for i(S _m (k, l)) and prove that if k + l = m then Havilands conjecture holds for the subset graph S _m (k, l). Furthermore, we give the exact value of (S _m (k, l)).This work was supported by National Natural Sciences Foundation of China (19871036).