Graph Labelings in Elementary Abelian 2-Groups

Abstract

Let $n\geq 2$ be an integer. We show that if $G$ is a graph such that every component of $G$ has order at least 3, and $|V(G)|\leq 2^n$ and $|V(G)|\neq 2^n-2$, then there exists an injective mapping $\varphi$ from $V(G)$ to an elementary abelian 2-group of order $2^n$ such that for every component $C$ of $G$, the sum of $\varphi(x)$ as $x$ ranges over $V(C)$ is $o$.