A Note on the Sparing Number of the Sieve Graphs of Certain Graphs

TL;DR

This paper introduces the concept of $k$-sieve graphs and investigates their sparing numbers, which relate to the minimum edges labeled with singleton sets needed for weak IASIs.

Contribution

It defines $k$-sieve graphs and analyzes their sparing numbers, extending the theory of integer additive set-indexers.

Findings

Defined $k$-sieve graphs of a given graph.

Derived bounds and formulas for sparing numbers of $k$-sieve graphs.

Established relationships between graph properties and sparing numbers.

Abstract

Let $\mathbb{N}_0$ denote the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ be its power set. An integer additive set-indexer (IASI) of a given graph $G$ is 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 $f^+ (uv) = f(u)+ f(v)$ is also injective. An IASI $f$ of a graph $G$ is said to be a weak IASI of $G$ if $|f^+(uv)|=\max(|f(u)|,|f(v)|)$ for all $u,v\in V(G)$. A graph which admits a weak IASI may be called a weak IASI graph. The sparing number of a graph $G$ is the minimum number of edges with singleton set-labels required for a graph $G$ to admit a weak IASI. In this paper, we introduce the notion of $k$-sieve graphs of a given graph and study their sparing numbers.