L(3,2,1)-labeling problem of square of path

Abstract

There are many problems in graph theory, where labeling of graphs is the only alternative to solve it. Graph labeling widely appeared in frequency assignment, communication network addressing, circuit design, X-ray crystallography, coding theory, missile guidance, signal processing, etc. For any graph [Formula: see text], where V is the node set and [Formula: see text] is the distance between nodes p and q, the [Formula: see text]-labeling of G is a mapping [Formula: see text] such that [Formula: see text] if [Formula: see text], [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text]. This paper is devoted to [Formula: see text]-labeling of squares of path (SOP) [Formula: see text] and we obtained unique results. This is the first result about [Formula: see text]-labeling of SOP. To the best of our knowledge, no result is available related to [Formula: see text]-labeling of square of any classes of graphs.