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## Proyecciones (Antofagasta)

##
*Print version* ISSN 0716-0917

### Proyecciones (Antofagasta) vol.33 no.4 Antofagasta Dec. 2014

#### http://dx.doi.org/10.4067/S0716-09172014000400002

**L(1,1)-Labeling of Direct Product of any Path and cycle**

**Deborah Olayide Ajayi **

*University of Ibadan *

**Charles Adefokun **

*Crawford University*

* Nigeria *

**ABSTRACT**

*Suppose that* [n] = {0, 1, 2,...,n} *is a set of non-negative integers and h,k G [n].The L (h,* k)*-labeling of graph G is the function l* : V(G) — [n] *such that* |l(u) — l(v)| > h *if the distance d(u,v) between u and v is* 1 *and* |l(u) — l(v)| > k *if* d(u,v) = 2. *Let L(V*(G)) = {l(v): v G V(G)} *and let p be the maximum value of L(V*(G)). *Then p is called X ^{i}^—number of G if p is the least possible member of* [n]

*such that G maintains an L(h,*k) —

*labeling. In this paper, we establish*X} —

*numbers of P*X

_{m}*P*X

_{n}and P_{m}*C*> 2.

_{n}graphs for all m,n**Keywords :** *L(1,1)-labeling, D-2 Coloring, Direct Product of Graphs, Cross Product ofGraphs, Path and Cycle.*

**Mathematics Subject Classification :** *Primary: 05C78.*

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**Deborah Olayide Ajayi**

Department of Mathematics University of Ibadan, Ibadan, Nigeria

e-mail : olayide.ajayi@mail.ui.edu.ng

388

*Deborah Olayide Ajayi and Charles Adefokun*

and

**Tayo Charles Adefokun**

Department of Computer and Mathematical Sciences,

Crawford University,

Nigeria

e-mail : tayoadefokun@crawforduniversity.edu.ng

**Received : November 2013. Accepted : September 2014**