Let $G=(V,E)$ be a connected simple graph‎. ‎A labeling $f‎: ‎Vto Z_2$ induces two edge labelings $f^+‎, ‎f^*‎: ‎E to‎ ‎Z_2$ defined by $f^+(xy) = f(x)+f(y)$ and $f^*(xy) =‎ ‎f(x)f(y)$ for each $xy in E$‎. ‎For $i in Z_2$‎, ...