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

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For $ u\in V(G) $,

$ N_V(u) $ = $ \left\lbrace w\in V(G) |uw \in E(G) \right\rbrace $ and

$ N_E(u) $ = $ \lbrace e\in E(G) | e= uv, \text { for some} $ $ v \in V(G) \rbrace $.

A bijective function

$ f: V(G)\cup E(G)\rightarrow \left\lbrace 1,2,3,\dots,|V(G) \cup E(G)| \right\rbrace $ is said to be a vertex-edge neighborhood prime labeling, if

for $ u \in V(G)$ with $deg(u)$ = 1$, $ gcd $ \left\lbrace f(w), f(uw)| w\in N_V(u) \right\rbrace = 1$ ;

for $ u \in V(G)$ with $deg(u)>1$, $ gcd \left\lbrace f(w)| w\in N_V(u) \right\rbrace = 1$ and $ gcd \left\lbrace f(e)| e\in N_E(u) \right\rbrace = 1$. A graph which admits a vertex-edge neighborhood prime labeling is called a vertex-edge neighborhood prime graph.

In this paper we prove $K_{m,n} \odot K_1$, $W_{n} \odot K_1$, $H_{n} \odot K_1$, $F_{n} \odot K_1$ and $S(K_{1,n}) \odot K_1$ are vertex-edge neighborhood prime graphs.