Edge-sum distinguishing labeling

Abstract

We study \emph{edge-sum distinguishing labeling}, a type of labeling recently introduced by Tuza in [Zs. Tuza, \textit{Electronic Notes in Discrete Mathematics} 60, (2017), 61-68] in context of labeling games. An \emph{ESD labeling} of an $n$-vertex graph $G$ is an injective mapping of integers $1$ to $l$ to its vertices such that for every edge, the sum of the integers on its endpoints is unique. If $l$ equals to $n$, we speak about a \emph{canonical ESD labeling}. We focus primarily on structural properties of this labeling and show for several classes of graphs if they have or do not have a canonical ESD labeling. As an application we show some implications of these results for games based on ESD labeling. We also observe that ESD labeling is closely connected to the well-known notion of \emph{magic} and \emph{antimagic} labelings, to the \emph{Sidon sequences} and to \emph{harmonious labelings}.