The Sparing Number of the Cartesian Products of Certain Graphs

TL;DR

This paper investigates the sparing number, a graph parameter related to integer additive set-indexers, specifically focusing on the Cartesian product of certain graphs and their properties.

Contribution

It introduces the concept of the sparing number for Cartesian products of graphs and explores its properties and bounds in this context.

Findings

Determined the sparing number for specific classes of Cartesian product graphs.

Established bounds and conditions for the sparing number in Cartesian products.

Extended the theory of integer additive set-indexers to product graphs.

Abstract

Let $\mathbb{N}_0$ be the set of all non-negative integers. An integer additive set-indexer (IASI) is defined as an injective function $f:V(G)\rightarrow \mathcal{P}(\mathbb{N}_0)$ such that the induced function $f^+:E(G) \rightarrow \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$ and $\mathcal{P}(\mathbb{N}_0)$ is the power set of $\mathbb{N}_0$. If $f^+(uv)=k \forall ~ uv\in E(G)$, then $f$ is said to be a $k$-uniform integer additive set-indexer. An integer additive set-indexer $f$ is said to be a weak integer additive set-indexer if $|f^+(uv)|=max(|f(u)|,|f(v)|) \forall ~ uv\in E(G)$. In this paper, we study about the sparing number of the cartesian product of two graphs.