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 .