A generalized shackle of any graph H admits a super H-antimagic total labeling
Dafik Dafik, Moh. Hasan, Yuli Nur Azizah, Ika Hesti Agustin · Journal of Physics Conference Series · 2017
Let H be a simple and connected graph. A shackle of graph H , denoted by G = shack ( H , v , n ), is a graph G constructed by non-trivial graphs H 1 , H 2 , ..., H n such that, for every 1 ≤ s , t ≤ n , H s and H t have no a common vertex with | s − t | ≥ 2 and for every 1 ≤ i ≤ n − 1, H i and H i +1 share exactly one common vertex v , called connecting vertex , and those k − 1 connecting vertices are all distinct. By a generalized shackle of graph, we mean the graph G = shack ( H , v , n ) by replacing the connecting vertex by any subgraph K ⊂ H and we denote such a graph as G = shack ( H , K ⊂ H , n ). Graph G = gshack ( H , K ⊂ H , n ) 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 an ( a , d )− H -antimagic total graph if there exists a bijective function f : V ( G ) ⋃ E ( G ) → {1, 2, ...,| V ( G )| + | E ( G )|} such that for all subgraphs isomorphic to H , the total H -weights w ( H ) = Σ v ∈ V ( H ) f ( v ) + Σ e ∈ E ( H ) f ( e ) form an arithmetic sequence { a , a + d , a + 2 d , ..., a + ( n − 1) d }, where a and d are positive integers and n is the number of all subgraphs isomorphic to H . If such a function exists 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 a super ( a , d )− H antimagic total labeling of G = gshack ( H , e ∈ H , n ) by using a partition technique. The result shows that there exist a super( a , d )− H antimagic total labeling for almost feasible difference d up to the determined upper bound.