Topological Integer Additive Set-Graceful Graphs

TL;DR

This paper introduces and studies a new class of graph labelings called topological integer additive set-graceful labelings, combining topological and additive set-labeling concepts to explore their properties and applications.

Contribution

It defines the concept of topological integer additive set-graceful labelings and investigates their properties, expanding the theory of graph labelings with new combined constraints.

Findings

Characterization of topological IASL and IASGL graphs.

Conditions for the existence of such labelings on various graph classes.

Examples illustrating the application of these labelings.

Abstract

Let $\mathbb{N}_0$ denote the set of all non-negative integers and $X$ be any subset of $X$. Also denote the power set of $X$ by $\mathcal{P}(X)$. An integer additive set-labeling (IASL) of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(X)$ such that the induced function $f^+:E(G) \to \mathcal{P}(X)$ is defined by $f^+ (uv) = f(u)+ f(v)$, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. An IASL $f$ is said to be a topological IASL (Top-IASL) if $f(V(G))\cup \{\emptyset\}$ is a topology of the ground set $X$. An IASL is said to be an integer additive set-graceful labeling (IASGL) if for the induced edge-function $f^+$, $f^+(E(G))= \mathcal{P}(X)-\{\emptyset, \{0\}\}$. In this paper, we study certain types of IASL of a given graph $G$, which is a topological integer additive set-labeling as well as an integer additive set-graceful labeling of $G$.