K-Product Cordial Labeling of Powers of Paths

Abstract

Let f be a map from V (G) to {0, 1, ..., k − 1}, where k is an integer and 1 ≤ k ≤ |V (G)|. For each edge uv assign the label f(u)f(v)(mod k). f is called a k-product cordial labeling if |v_f (i) − v_f (j)| ≤ 1, and |e_f (i) − e_f (j)| ≤ 1, i, j ∈ {0, 1, ..., k − 1}, where v_f (x) and e_f (x) denote the number of vertices and edges, respectively labeled with x (x = 0, 1, ..., k − 1). In this paper, we add some new results on k-product cordial labeling and prove that the graph P²_n is 4-product cordial. Further, we study the k-product cordial behaviour of powers of paths P³_n, P⁴_n and P⁵_n for k = 3 and 4.

Key words and phrases. cordial labeling, product cordial labeling, k-product cordial labeling, 3-product cordial labeling, 4-product cordial labeling.

2010 Mathematics Subject Classification. 05C78.