Fans are cycle-antimagic

A simple graph G = (V,E) admits an H-covering if every edge in E belongs at least to one subgraph of G isomorphic to a given graph H . Then the graph G admitting an H-covering is (a, d)-H-antimagic if there exists a bijection f : V ∪ E → {1, 2, . . . , |V | + |E|} such that, for all subgraphs H ′ of G isomorphic to H , the H ′-weights, wtf(H ′) = ∑ v∈V (H′) f(v) + ∑ e∈E(H′) f(e), form an arithmetic progression with the initial term a and the common difference d. Such a labeling is called super if the smallest possible labels appear on the vertices. This paper is devoted to studying the existence of super (a, d)-Hantimagic labelings for fans when subgraphs H are cycles.