DECOMPOSITION OF COMPLETE GRAPHS AND COMPLETE BIPARTITE GRAPHS INTO COPIES OF P sub n sub sup 3 sup or S sub 2 sub P sub n sub sup 3 sup AND HARMONIOUS LABELING OF K sub 2 sub + P sub n sub | Selvaraju | Journal of the Indonesian Mathematical

J. Indones. Math. Soc.

Special Edition (2011), pp. 109–122.

DECOMPOSITION OF COMPLETE GRAPHS AND

COMPLETE BIPARTITE GRAPHS INTO COPIES OF

P

3

n

OR

S

₂

₍

P

3

n

)

AND HARMONIOUS LABELING OF

K

2

+

P

n

P. Selvaraju

1

and G. Sethuraman

2

1

Department of Mathematics, VEL TECH, Avadi, Chennai-600062, India,

pselvar@yahoo.com

2

Department of Mathematics, Anna University, Chennai-600025, Chennai,

India, sethu@annauniv.edu

Abstract. _{In this paper, the graphs P}3

n and S2(P

3

n) are shown to admit an α -valuation, whereP3

nis the graph obtained from the pathPnby joining all the pairs of verticesu, vofPnwithd(u, v) = 3 andS2(P_n3) is the graph obtained fromP_n3by

merging the centre of the starSn1 and that of the starSn2 respectively at the two

unique 2-degree vertex ofPn3(the origin and terminus of the pathPncontained in P3

n). It follows from the significant theorems due to Rosa [1967] and EI-Zanati and Vanden Eynden [1996] that the complete graphsK2cq+1or the complete bipartite

graphsKmq,nq can be cyclically decomposed into the copies of P

3

n or copies of S2(Pn3), wherec, m, nare arbitrary positive integer and qdenotes either|E(P

3

n)| or|E(S2(Pn3))|. Further, it is shown that join of complete graphK2 and pathPn, denotedK2+Pn, forn≥1 is harmonious graph.

Key words:α-labeling, harmonious labeling,P3

ngraphs, join, path.

2000 Mathematics Subject Classification: 05C78.