### L(1,1,1)-labeling of path, bouquet of cycles and sun graph

#### Abstract

For a given graph G(V,E), L(1,1,1)-labeling problem is an assignment from vertex set V to the set of non negative integers. If Z^+ be the non negative integers then L(1,1,1)-labeling is a function f:V→Z^+ such that for any two vertices x and y, |f(x)-f(y)|≥1, when d(x,y)=1; |f(x)-f(y)|≥1, when d(x,y)=2; and |f(x)f(y)|≥1, when d(x,y)=3. The L(1,1,1)-chromatic number λ_1,1,1 is the smallest positive integer such that G has an L(1,1,1)-labeling with λ_1,1,1 as the maximum label. In this paper we determine the L(1,1,1)-chromatic number for a path, a cycle, bouquet of cycles joining at a vertex (all are of finite lengths) and sun graph. We also present a lower and upper bounds for λ_1,1,1 in terms of the maximum degree of G.

**Published:**2020-05-26

**How to Cite this Article:**Nasreen Khan, L(1,1,1)-labeling of path, bouquet of cycles and sun graph, J. Math. Comput. Sci., 10 (2020), 1360-1374 Copyright © 2020 Nasreen Khan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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