On distance labelings of 2-regular graphs

Abstract

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 w ψ ( v ) = ∑ u ∈ N ( v ) ψ( u ). The function ψ is called a distance magic labeling of G , if w ψ ( 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 +2 d , ... , 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.