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 H such that, for every 1 · s; t · n, H have no a common vertex with js ¡ tj ¸ 2 and for every 1 · i · n ¡ 1, H s i and H and H t 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 ¤ i+1 )-H-antimagic total graph of second order if there exist a bijective function f : V (G) [E(G) ! f1; 2; : : : ; jV (G)j +jE(G)jg such that for all subgraphs isomorphic to H, the total H-weights W(H) = P v2V (H) f(v) + P f(e) form an arithmetic sequence of second order of fa ¤ ; a ¤ +d ¤ ; a ¤ +3d ¤ ; a ¤ +6d ¤ ; : : : ; a e2E(H) ¤ +( n 2 ¡n 2 )d ¤ g, 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.