On the Roman Edge Domination Number of a Graph

For an integer n ≥ 2, let I ⊂ {0, 1, 2, · · · , n}. A Smarandachely Roman sdominating function for an integer s, 2 ≤ s ≤ n on a graph G = (V,E) is a function f : V → {0, 1, 2, · · · , n} satisfying the condition that |f(u)− f(v)| ≥ s for each edge uv ∈ E with f(u) or f(v) ∈ I . Similarly, a Smarandachely Roman edge s-dominating function for an integer s, 2 ≤ s ≤ n on a graph G = (V,E) is a function f : E → {0, 1, 2, · · · , n} satisfying the condition that |f(e) − f(h)| ≥ s for adjacent edges e, h ∈ E with f(e) or f(h) ∈ I . Particularly, if we choose n = s = 2 and I = {0}, such a Smarandachely Roman sdominating function or Smarandachely Roman edge s-dominating function is called Roman dominating function or Roman edge dominating function. The Roman edge domination number γre(G) of G is the minimum of f(E) = ∑ e∈E f(e) over such functions. In this paper we first show that for any connected graph G of q ≥ 3, γre(G) + γe(G)/2 ≤ q and γre(G) ≤ 4q/5, where γe(G) is the edge domination number of G. Also we prove that for any γre(G)-function f = {E0, E1, E2} of a connected graph G of q ≥ 3, |E0| ≥ q/5 + 1, |E1| ≤ 4q/5− 2 and |E2| ≤ 2q/5.