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
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Link to Publisher's Edition
Lam, P., Shiu, W., & Sun, L. (1999). On independent domination number of regular graphs. Discrete Mathematics, 202 (1-3). https://doi.org/10.1016/S0012-365X(98)00350-1