A Study on the Modular Sumset Labeling of Graphs

TL;DR

This paper explores the properties and characteristics of modular sumset labelings of graphs, where vertices are labeled with subsets of integers modulo n, and examines how these labelings induce injective functions on edges.

Contribution

It introduces and analyzes the concept of modular sumset labelings and indexers of graphs, providing foundational properties and characteristics of these labelings.

Findings

Characterization of modular sumset labelings

Conditions for injectivity of labelings and induced functions

Properties of sumset indexers in graphs

Abstract

For a positive integer $n$, let $\mZ$ be the set of all non-negative integers modulo $n$ and $\sP(\mZ)$ be its power set. A modular sumset valuation or a modular sumset labeling of a given graph $G$ is an injective function $f:V(G) \to \sP(\mZ)$ such that the induced function $f^+:E(G) \to \sP(\mZ)$ defined by $f^+ (uv) = f(u)+ f(v)$. A sumset indexer of a graph $G$ is an injective sumset valued function $f:V(G) \to \sP(\mZ)$ such that the induced function $f^+:E(G) \to \sP(\mZ)$ is also injective. In this paper, some properties and characteristics of this type of modular sumset labeling of graphs are being studied.