L (3, 1, 1)-labeling numbers of square of paths, complete graphs and complete bipartite graphs
Sk Amanathulla, Sankar Sahoo, Madhumangal Pal · Journal of Intelligent & Fuzzy Systems · 2018
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 ( P n 2 ) , square of complete graphs ( K n 2 ) and square of complete bipartite graphs ( K p , q 2 ) 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.