The Connected and Disjoint Union of Semi Jahangir Graphs Admit a Cycle-Super (a, d)-Atimagic Total Labeling

Abstract

We assume that all graphs in this paper are finite, undirected and no loop and multiple edges. Given a graph G of order p and size q.LetH

,H be subgraphs of G.ByH -covering, we mean every edge in E(G) belongs to at least one subgraph of G isomorphic to a given graph H.AgraphG is said to be an (a, d)-H-antimagic total labeling if there exist a bijective function f : V (G)∪E(G) →{1, 2,...,p+q} such that for all subgraphs H isomorphic to H, the total H-weights w(H)=

v∈V (H

) f(v)+

e∈E(H

)

f(e) form an arithmetic sequence {a, a+d, a+2d, ..., a+(s−1)d}, where a and d are positive integers and s is the number of all subgraphs H isomorphic to H.Sucha labeling is called super if f : V (G) →{1, 2,...,|V (G)|}. In this paper, we will discuss a cycle-super (a,d)-atimagicness of a connected and disjoint union of semi jahangir graphs. The results show that those graphs admit a cycle-super (a,d)-atimagic total labeling for some feasible d ∈{0, 1, 2, 4, 6, 7, 10, 13, 14}.