Edge-sum distinguishing labeling

TL;DR

This paper investigates edge-sum distinguishing (ESD) labelings of graphs, exploring their structural properties, existence conditions for canonical labelings, and connections to other graph labelings, with implications for related labeling games.

Contribution

It characterizes when various graph classes admit canonical ESD labelings and links ESD labelings to magic, antimagic, Sidon, and harmonious labelings.

Findings

Certain graph classes have or lack canonical ESD labelings.

Connections established between ESD labelings and other well-known labelings.

Implications for ESD-based games derived from structural properties.

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}.