Smarandachely k-Constrained Number of Paths and Cycles

A Smarandachely k-constrained labeling of a graph G(V, E) is a bijective mapping f : V ∪ E → {1, 2, .., |V | + |E|} with the additional conditions that |f(u) − f(v)| ≥ k whenever uv ∈ E, |f(u)−f(uv)| ≥ k and |f(uv)−f(vw)| ≥ k whenever u 6= w, for an integer k ≥ 2. A graph G which admits a such labeling is called a Smarandachely k-constrained total graph, abbreviated as k−CTG. The minimum number of isolated vertices required for a given graph G to make the resultant graph a k −CTG is called the k-constrained number of the graph G and is denoted by tk(G). In this paper we settle the open problems 3.4 and 3.6 in [4] by showing that tk(Pn) = 0, if k ≤ k0; 2(k − k0), if k > k0 and 2n ≡ 1 or 2 (mod 3); 2(k − k0) − 1 if k > k0; 2n ≡ 0(mod 3) and tk(Cn) = 0, if k ≤ k0; 2(k − k0), if k > k0 and 2n ≡ 0 (mod 3); 3(k − k0) if k > k0 and 2n ≡ 1 or 2 (mod 3), where k0 = ⌊ 2n−1 3 ⌋.