Super (a*, d*)-ℋ-antimagic total covering of second order of shackle graphs

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. The graph G is said to be an (a*, d*)-H-antimagic total graph of second order if there exist 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 form an arithmetic sequence of second order of , where a* and d* are positive integers and n is the number of all subgraphs isomorphic to H. An (a*, d*)-H-antimagic total labeling of second order f is called super if the smallest labels appear in the vertices. In this paper, we study a super (a*, d*)-H antimagic total labeling of second order of G = shack(H, v, n) by using a partition technique of second order.