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

Deborah Olayide Ajayi

University of Ibadan

Charles Adefokun

Crawford University

Nigeria

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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 Xⁱ^—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_m X P_n and P_m X C_n graphs for all m,n > 2.

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