On Disjunctive and Conjunctive Set-Labelings of Graphs

TL;DR

This paper introduces and studies two new types of set-labelings for graphs, called conjunctive and disjunctive set-labelings, exploring their properties and characteristics.

Contribution

It defines conjunctive and disjunctive set-labelings of graphs and investigates their properties, expanding the framework of graph labelings.

Findings

Defined conjunctive and disjunctive set-labelings.

Analyzed properties and characteristics of these labelings.

Established foundational results for future research.

Abstract

Let $X$ be a non-empty set and $\sP(X)$ be its power set. A set-valuation or a set-labeling of a given graph $G$ is an injective function $f:V(G) \to \sP(X)$ such that the induced function $f^{\ast}:E(G) \to \sP(X)$ defined by $f^{\ast} (uv) = f(u)\ast f(v)$, where $\ast$ is a binary operation on sets. A set-indexer of a graph $G$ is an injective set-valued function $f:V(G) \to \sP(X)$ such that the induced function $f^{\ast}:E(G) \to \sP(X)$ is also injective. In this paper, two types of set-labelings, called conjunctive set-labeling and disjunctive set-labeling, of graphs are introduced and some properties and characteristics of these types of set-labelings of graphs are studied.