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3 It seems a hard problem to determine in general the full range of values of k and d and possible partitions (k1 , k2 ) of k, 0 ≤ k1 < k2 , with k2 − k1 = rd and r < a such that Ca,b , 2 ≤ a ≤ b, has a (k, d)-arithmetic numbering f for which k1 , k2 ∈ f (Ca,b ). In particular, if (a, b, k, d)=(a, b, a, 1) then for the partition (0, a) of a, since 1 | (a − 0) and r=a, we see that Ca,b a ≤ b, has an (a, 1)-arithmetic numbering f with 0, a ∈ f (Ca,b ). 7. 7 Every caterpillar is sequential. 2, any balanced bipartite graph G with {A, B}, | A |= a ≤ b =| B |, has a (k, d)-arithmetic numbering f with k1 , k2 ∈ f (G).

Let P be a parallel transformation of T that contains P1 as one of the constituent ept’s. Since vi+t vj−t is an edge in the path P(T) it follows that i+t+1=j-t ⇒ j = i + 2t + 1. The value of the edge vi vj is f + (vi vj ) = f + (vi vi+2t+1 ) = f (vi ) + f (vi+2t+1 ) If i is odd and 1 ≤ i ≤ n, then d + k + (q − 1)d + f (vi ) + f (vi+2t+1 ) = (i−2) 2 (i+2t+1−2) 2 (5) d = k + (q − 1)d + (i + t − 1)d. If i is even and 2 ≤ i ≤ n, then f (vi ) + f (vi+2t+1 ) = k + (q − 1)d + (i−2) 2 d+ (i+2t+1−1) 2 = k + (q − 1)d + (i + t − 1)d 35 (6) d (7) Therefore, from (5), (6), (7), we get f + (vi vj ) = k + (q − 1)d + (i + t − 1)d ∀i, 1 ≤ i ≤ n (8) The value of the edge vi+t vj−t is f + (vi+t vj−t ) = f (vi+t ) + f (vj−t ) = f (vi+t ) + f (vi+t+1 ) If i+t is odd, then f (vi+t ) + f (vi+t+1 ) = (i+t−1) 2 d + k + (q − 1)d + (i+t+1−2) 2 d = k + (q − 1)d + (i + t − 1)d If i+t is even, then f (vi+t ) + f (vi+t+1 ) = (i+t+1−1) 2 d + k + (q − 1)d + (i+t−2) 2 (9) (10) d = k + (q − 1)d + (i + t − 1)d (11) Therefore, from (9), (10), (11), we get f + (vi+t vj−t ) = k + (q − 1)d + (i + t − 1)d (12) From (8) and(12), f + (vi vj ) = f + (vi+t vj−t ).

Combinatorics, 10, 2007, DS6. M and Shetty, Sudhakar, On Arithmetic Graphs, Indian J. , 33(8), 2002, 1275-1283.