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In this paper, the graphs P 3 n and S2(P 3 n) are shown to admit an αvaluation, where P 3 n is the graph obtained from the path Pn by joining all the pairs of vertices u, v of Pn with d(u, v) = 3 and S2(P 3 n) is the graph obtained from P 3 n by merging the centre of the star Sn1 and that of the star Sn2 respectively at the two unique 2-degree vertex of Pn3 (the origin and terminus of the path Pn contained in P 3 n). It follows from the significant theorems due to Rosa [1967] and EI-Zanati and Vanden Eynden [1996] that the complete graphs K2cq+1 or the complete bipartite graphs Kmq,nq can be cyclically decomposed into the copies of P 3 n or copies of S2(P 3 n), where c,m, n are arbitrary positive integer and q denotes either |E(P 3 n)| or |E(S2(P 3 n))|. Further, it is shown that join of complete graph K2 and path Pn, denoted K2 + Pn, for n ≥ 1 is harmonious graph.

journal of the indonesian mathematical society

DECOMPOSITION OF COMPLETE GRAPHS AND COMPLETE BIPARTITE GRAPHS INTO COPIES OF P&lt;sub&gt;n&lt;/sub&gt;&lt;sup&gt;3&lt;/sup&gt; or S&lt;sub&gt;2&lt;/sub&gt;P&lt;sub&gt;n&lt;/sub&gt;&lt;sup&gt;3&lt;/sup&gt; AND HARMONIOUS LABELING OF K&lt;sub&gt;2&lt;/sub&gt;+ P&lt;sub&gt;n&lt;/sub&gt;