A Study on the Product Set-Labeling of Graphs

TL;DR

This paper introduces and explores product set-labeling of graphs, a new set-labeling concept where vertex labels are sets of positive integers and edge labels are their product sets, analyzing properties of graphs that admit such labelings.

Contribution

It proposes the novel notion of product set-labeling of graphs and investigates the properties of graphs that can be labeled in this manner.

Findings

Defined product set-labeling for graphs with labels as sets of positive integers.

Characterized classes of graphs that admit product set-labelings.

Analyzed properties and conditions for the existence of such labelings.

Abstract

Let $X$ be a non-empty ground set and $\mathscr{P}(X)$ be its power set. A set-labeling (or a set-valuation) of a graph $G$ is an injective set-valued function $f:V(G)\to \mathscr{P}(X)$ such that the induced function $f^*:E(G) \to \mathscr{P}(X)$ is defined by $f^*(uv)=f(u)\ast f(v)$, where $f(u)\ast f(v)$ is a binary operation of the sets $f(u)$ and $f(v)$. A graph which admits a set-labeling is known to be a set-labeled graph. A set-labeling $f$ of a graph $G$ is said to be a set-indexer of $G$ if the associated function $f^*$ is also injective. In this paper, we introduce a new notion namely product set-labeling of graphs as an injective set-valued function $f:V(G)\to \mathscr{P}(\mathbb{N})$ such that the induced edge-function $f^*:V(G)\to \mathscr{P}(\mathbb{N})$ is defined as $^*f(uv)=f(u)\ast f(v) \forall\ uv\in E(G)$, where $f(u)\ast f(v)$ is the product set of the set-labels $f(u)$ and $f(v)$, where $\mathbb{N}$ is the set of all positive integers and discuss certain properties of the graphs which admit this type of set-labeling.