A Study on the Product Set-Labeling of Graphs

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.