Department of Mathematics
Let G be a simple graph. The independent domination number i(G) is the minimum cardinality among all maximal independent sets of G. Haviland (1995) conjectured that any connected regular graph G of order n and degree δ ⩽ n/2 satisfies i(G) ⩽ ⌈2n/3δ⌉δ/2. In this paper, we will settle the conjecture of Haviland in the negative by constructing counterexamples. Therefore a larger upper bound is expected. We will also show that a connected cubic graph G of order n ⩾ 8 satisfies i(G) ⩽ 2n/5, providing a new upper bound for cubic graphs.
Independent domination number, Regular graph
Source Publication Title
Link to Publisher's Edition
Lam, Peter Che Bor, Wai Chee Shiu, and Liang Sun. "On independent domination number of regular graphs." Discrete Mathematics 202.1-3 (1999): 135-144.