An upper bound on triple Roman domination

For a graph G = (V, E), a triple Roman dominating function (3RDfunction) is a function f : V → {0, 1, 2, 3, 4} having the property that (i) if f(v) = 0 then v must have either one neighbor u with f(u) = 4, or two neighbors u, w with f(u) + f(w) ≥ 5 or three neighbors u, w, z with f(u) = f(w) = f(z) = 2, (ii) if f(v) = 1 then v must have one neighbor u with f(u) ≥ 3 or two neighbors u, w with f(u) = f(w) = 2, and (iii) if f(v) = 2 then v must have one neighbor u with f(u) ≥ 2. The weight of a 3RDF f is the sum f(V ) = ∑ v∈V f(v), and the minimum weight of a 3RD-function on G is the triple Roman domination number of G, denoted by γ[3R] (G). In this paper, we prove that for any connected graph G of order n with minimum degree at least two, γ[3R] (G) ≤ 3n 2 .