VERTEX-EDGE NEIGHBORHOOD PRIME LABELING IN THE CONTEXT OF CORONA PRODUCT

Abstract

Let G be a graph with vertex set V (G) and edge set E(G).  For u ∈ V (G), N_V (u) = {w ∈ V (G)|uw ∈ E(G)} and N_E(u) = {e ∈ E(G)|e = uv,  for some  {| ∈  } { | ∈  } v ∈ V (G)}. A bijective function f : V (G) ∪ E(G) → {1, 2, 3, . . . , |V (G) ∪ E(G)|} is said to be a vertex-edge neighborhood prime labeling, if for u ∈ V (G) with deg(u) = 1, gcd {f (w), f (uw)|w ∈ N_V (u)} = 1 ;  for u ∈ V (G) with deg(u) > 1, gcd   f (w) w     N_V (u)   = 1 and gcd   f (e) e     N_E(u)   = 1.  A graph which admits a vertex-edge neighborhood prime labeling is called a vertex-edge neighborhood prime graph. In this paper we prove K_m,n Ⓢ K₁, W_n Ⓢ K₁, H_n Ⓢ K₁, F_n Ⓢ K₁ and S(K_1,n) Ⓢ K₁ are vertex-edge neighborhood prime graphs.