A generalized shackle of any graph H admits a super H-antimagic total labeling

Abstract

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 H1, H2, …, Hn such that, for every 1 ≤ s, t ≤ n, Hs and Ht have no a common vertex with |s − t| ≥ 2 and for every 1 ≤ i ≤ n − 1, Hi and Hi+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) = Σv∈V(H) f(v) + Σe∈E(H) f(e) form an arithmetic sequence {a, a + d, a + 2d, …, 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.