Cyclic-antimagic construction of ladders

A simple graph G = (V, E) admits an H-covering if every edge in the edge set E(G) belongs to at least one subgraph of G isomorphic to a given graph H. A graph G having an H-covering is called (a, d)-H-antimagic if the function h : V(G) ∪ E(G)→ {1, 2, . . . , |V(G)|+ |E(G)|} defines a bijective map such that, for all subgraphs H′ of G isomorphic to H, the sums of labels of all vertices and edges belonging to H′ constitute an arithmetic progression with the initial term a and the common difference d. Such a graph is named as super (a, d)-H-antimagic if h(V(G)) = {1, 2, 3, . . . , |V(G)|}. For d = 0, the super (a, d)-H-antimagic graph is called H-supermagic. In the present paper, we study the existence of super (a, d)-cycle-antimagic labelings of ladder graphs for certain differences d.