# L(2, 2, 1)-labeling problems of square of paths

### Abstract

L(p, q)-labeling problem is a well studied problem in the last three decades for its wide application, specially in frequency assignment in (mobile) communication system, Xray crystallography, coding theory, radar, astronomy, circuit design etc. L(2, 2, 1)- labeling is an extension of L(p, q)-labeling is now becomes a well studied problem due to its application. Motivated from this point of view, we consider L(2, 2, 1)-labeling problem for square of paths. Let G = (V, E) be a graph. The L(2, 2, 1)-labeling of the graph G is a mapping η : V → {0, 1, 2, . . .} so that |η(x) − η(y)| ≥ 2 if d(x, y) = 1 or 2, |η(x) − η(y)| ≥ 1 if d(x, y) = 3, where V is the vertex set and d(x, y) is the distance (i.e. minimum number of edges in the shortest path between x and y) between the vertices x and y. λ2,2,1(G) is the L(2, 2, 1)-labeling number of G, which is the the largest non-negative integer which is used to label the graph G. In labeling problems of graph the main target is to find the exact value of λ2,2,1(G) or to minimize it. In this paper we have studied L(2, 2, 1)-labeling of squares of paths and it is proved that λ2,2,1(P 2 n )

0 if n = 1 2 if n = 2 4 if n = 3 6 if n = 4 8 if n = 5 9 if n = 6, 7, 8, 9, 10 10 if n ≥ 11 Also a labeling procedure is presented to label a square of paths. The result of this paper is exact and also it is unique. This is the first result about L(2, 2, 1)-labeling of square of paths

**Current Science**, [S.l.], v. 113, n. Issue 7, dec. 2017. ISSN 0011-3891. Available at: <http://currentscience.org/index.php/CS/article/view/193>. Date accessed: 20 apr. 2018.

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