The Sparing Number of the Cartesian Products of Certain Graphs

Let ${\mathbb{N}}_0$ be the set of all non-negative integers. An integer additive set-indexer (IASI) is defined as an injective function $f:V(G)\to \mathcal{P}({\mathbb{N}}_0)$ such that the induced function $f^{+}:E(G)\to \mathcal{P}({\mathbb{N}}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$isalsoinjective,where$f(u)+f(v)$isthesumsetof$f(u)$and$f(v)$and$$\mathcal{P}$($\mathbb{N}$_0)$isthepowersetof$$\mathbb{N}$_0$\mathrm{.}If$f^+(uv)=k \forall ~ uv\in E(G)$,then$f$issaidtobea$k$-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.