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A 2-rainbow dominating function of a graph G is a function g that assigns to each vertex a set of colors chosen from the set {1, 2} so that for each vertex with g(v) = ∅ we have u∈N (v) g(u) = {1, 2}. The minimum of g(V (G)) = v∈V (G) |g(v)| over all such functions is called the 2-rainbow domination number γ 2r (G). A Roman dominating function on a graph G… (More)

Let G = (V, E) be a simple graph. A set D ⊆ V is a dominating set of G if every vertex of V − D is adjacent to a vertex of D. The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set of G. We prove that if G is a Hamiltonian graph of order n with minimum degree at least six, then γ(G) ≤ 6n 17 .

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