Even and odd strongly multiplicative graphs

Abstract

A graph $G=(V(G), E(G))$ with $p$ vertices is said to be even strongly multiplicative if the vertices of $G$ can be labeled with $p$ distinct integers $1,2, \ldots, p$ such that the labels induced on the edges by the product of labels of the end vertices are all distinct and even.
If the vertices of $G$ are labeled with distinct integers $1,2,3, \ldots, 2 p-1$ such that the label induced on the edges by the product of labels of the end vertices should all be odd and distinct then the graph is said to be odd strongly multiplicative.