Perfect Roman domination in regular graphs

A perfect Roman dominating function on a graph G is a function f: V (G) → (0, 1, 2) satisfying the condition that every vertex u with f(u) = 0 is adjacent to exactly one vertex v for which f(v) = 2. The weight of a perfect Roman dominating function f is the sum of the weights of the vertices. The perfect Roman domination number of G, denoted γ Rp(G), is the minimum weight of a perfect Roman dominating function in G. We show that if G is a cubic graph on n vertices, then γRp (G) ≤ 3/4n, and this bound is best possible. Further, we show that if G is a k-regular graph on n vertices with k at least 4, then γ Rp(G) ≤ (k2+k+3/k2+3k+1) n