Cycle-Super Antimagicness of Connected and Disconnected Tensor Product of Graphs

Abstract

Let H be a graph. Graph G = (V, E) admits a H-covering, if every edge in E(G) belongs to at least one subgraph of G isomorphic to a given graph H. A graph G is said to be an (a, d)-H-antimagic total graph if there exist a bijective function f : V(G) ∪ E(G) → {1, 2,. . ., |V(G)| + |E(G)|} such that for all subgraphs H1 isomorphic to H, the total H-weights w(H) = ∑v∈V(H1) f (v) + ∑e∈E(H1) f (e) form an arithmetic sequence {a, a + d, a + 2d, ..., a +(t − 1)d}, where a and d are positive integers and t is the number of all subgraphs H1 isomorphic to H. If such a function exist then f is called an (a, d)-H-antimagic total labeling of G. An (a, d)-H-antimagic total labeling f is called super if f : V(G) → {1, 2,. . ., |V(G)|}. In this paper, we study the super (a, d)-C2r −antimagic total labeling for a connected and disconnected tensor product of Cr and Pn, for odd r ≥ 3 and any n ≥ 3. The result shows that a tensor product of Cr and Pn and disjoint union of a tensor product of Cr and Pn, for odd r ≥ 3 and any n ≥ 3, admit a super(a, d)-C2r −antimagic total labeling for some feasible d.