A Study On (a,d)-Antimagic Graphs Using Partition

摘要

A connected graph G(V,E) is said to be (a,d)-antimagic if there exist positive integers a and d and a bijection f: E → {l,2,...,|E|} such that the induced mapping gf: V → N defined by gf(v) = ∑e∈I(v) f(e), where I(v) = {e∈E/ e is incident to v}, vs Vis injective and gf{V) = {a,a+d,a+2d,...,a+(|V|-l)d}. In this paper, using partition, we prove that (i) the 1-sided infinite path P1 is (l,2)-antimagic (ii) path P_2n+1 is (n,l)-antimagic and (iii) (n+2,l)-antimagic labeling is the unique (a,d)-antimagic labeling of C_2n+1 and graphs K1 + (K1∪ K_2), P_2n and C_2n are not (a,d)-antimagic. For a,deiV, on (a,d)-antimagic graph G, we obtain a new relation, a+(p-l)d < △(2q-△+l)/2. Using the results on (a,d)-antimagic labeling of C_2n and C_2n+1, we obtain results on the existence of (a,d)-arithmetic sequences of length 2n and 2n+l, respectively.