On distance labelings of 2-regular graphs

Let G be a graph with |V(G)| vertices and ψ : V(G) → {1, 2, 3, ... , |V(G)|} be a bijective function. The weight of a vertex v ∈ V(G) under ψ is wψ(v) = ∑u ∈ N(v)ψ(u). The function ψ is called a distance magic labeling of G, if wψ(v) is a constant for every v ∈ V(G). The function ψ is called an (a,d)-distance antimagic labeling of G, if the set of vertex weights is a, a+d, a+2d, ... , a+(|V(G)|-1)d. A graph that admits a distance magic (resp. an (a,d)-distance antimagic) labeling is called distance magic (resp. (a,d)-distance antimagic). In this paper, we characterize distance magic 2-regular graphs and (a,d)-distance antimagic some classes of 2-regular graphs.