On edge-prime cubic graphs with small components

Let G = G ( V , E ) be a ( p , q ) -graph. A bijection f : E → { 1 , 2 , 3 , … , q } is called an edge-prime labeling if for each edge u v in E , we have G C D ( f + ( u ) , f + ( v ) ) = 1 where f + ( u ) = ∑ u w ∈ E f ( u w ) . A graph that admits an edge-prime labeling is called an edge-prime graph. In this paper we obtained some sufficient conditions for graphs with regular component(s) to admit or not admit an edge-prime labeling. Consequently, we proved that if G is a cubic graph with every component is of order 4 , 6 or 8 , then G is edge-prime if and only if G ≆ K 4 or n K ( 3 , 3 ) , n ≡ 2 , 3 ( mod 4 ) . We conjectured that a connected cubic graph G is not edge-prime if and only if G ≅ K 4 .