Critical Sets in Edge-Magic Total Labelings

Abstract

Let $\lambda$ be an edge-magic total (EMT) labeling of graph $G(V,E)$. Let $W\subset V(G)\cup E(G)$. Any restriction of $\lambda$ to $W$ is called a $partialEMTlabeling$ on $G$. A partial EMT labeling $\pi$ is a critical set in $\lambda$ if $\lambda$ is the only edge-magic total labeling having $\pi$ as its partial EMT labeling, and no proper restriction of $\pi$ satisfies the first condition. In this paper, we study the property of critical sets in such a labeling. We determine critical sets in an EMT labeling for a given graph $G$.