L (3,1,1)-labeling numbers of square of paths,complete graphs and complete bipartite graphs

Abstract

For a graph G = (V, E) the L (3, 1, 1)-labeling is a mapping μ from the vertex set V to the set of non-negative integers {0, 1, 2, …} such that |μ (x) - μ (y) |≥3 if d (x, y) =3 and |μ (x) - μ (y) |≥1 if d (x, y) =2 or 3, where d (x, y) is the distance between the vertices x and y. λ_3,1,1 (G) represents the L (3, 1, 1)-labeling number of the graph G and it is the largest non-negative integer used to label the graph G.

In the present article we have studied L (3, 1, 1)-labeling of squares of some simple graphs, namely square of paths $\left(P_n^2\right)$ , square of complete graphs $\left(K_n^2\right)$ and square of complete bipartite graphs $\left(K_{p,q}^2\right)$ and we obtained good results for these graph classes. The results produced in this article are unique and exact. This is the first result about L (3, 1, 1)-labeling for these graph classes.