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2 Every Tp -tree T is (k+(q-1)d, d)-arithmetic for all positive integers k and d. Proof Let T be a Tp -tree with n+1 vertices. Pk ) of the ept’s Pi used to arrive at the path P(T). Clearly Ed and Ep have the same number of edges. Then denote the vertices of P(T) successively as v1 , v2 ,... vp starting from one pendent vertex of P(T) right up to other. Define f : V (P (T )) −→N by f (vi ) = (i−1)d 2 for odd i, k + (q − 1)d + (i−2) 2 1≤i≤n+1 d for even i, 2 ≤ i ≤ n + 1 where k and d are positive integers and q is the number of edges of T.

6 For any nonnegative integer r and positive integer d, C4t+3 is ((2t+1)d+2r, d)-arithmetic. Proof Under the hypotheses, the map f : V (C4t+3 ) −→ N defined by 46 f (ui ) = r + ( i−1 )d, if i is odd 2 1 (2t + 1)d + r + 2 id, if i is even (19) is a required arithmetic numbering of C4t+3 . This completes the proof. 8 by taking d=2 and r=0. 5, that for an odd cycle to be (k, d)-arithmetic it is necessary that k and d to be parity. This prompts us to the following: Conjecture 3 (1) If C4t+1 is (k, d)-arithmetic then k=2dt+2r for some integer r ≥ 0.

5, that for an odd cycle to be (k, d)-arithmetic it is necessary that k and d to be parity. This prompts us to the following: Conjecture 3 (1) If C4t+1 is (k, d)-arithmetic then k=2dt+2r for some integer r ≥ 0. (2) If C4t+3 is (k, d)-arithmetic then k=(2t+1)d+2r for some integer r ≥ 0. 47 REFERENCES [1] Acharya, B. D and Hegde, S. Graph Theory, 14(3), 1989, 275-299. [2] Balakrishnan, R and Ranganathan, K, A Text Book of Graph Theory, Springer, Newyork, 1996. [3] Chartrand, Gray and Lesniak, Linda, Graphs and Digraphs, Chapman and Hall, London, 1996.

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