Vertex-graceful graphs

Abstract

A (p, q)-graph G = (V, E) is called vertex-graceful if it admits a vertex-graceful numbering, which is defined as an injection f : E → {0, 1, 2,…, q*}, q* = max{p, q} such that the function fV : V → ℕ defined by the rule fV (v) = max{f (e) : e ∈ Ev and v ∈ e}.-min {f(e) : e ∈ Ev and v ∈ e} satisfies the property that fV (V) ≔ {fV (u) : u ∈ V} = {1, 2,…, p}, where Ev denotes the set of edges in G that are incident at v and ℕ denotes the set of natural numbers. A study of this new notion is the prime objective of this paper.