A Note on Sparing Number Algorithm of Graphs

TL;DR

This paper introduces an algorithm to determine the sparing number of arbitrary graphs based on weak integer additive set-labelings, which involves assigning set labels to vertices and edges with specific properties.

Contribution

The paper presents a novel algorithm for calculating the sparing number in graphs using weak integer additive set-labelings, expanding understanding of graph labelings.

Findings

Algorithm effectively computes sparing number for various graphs.

Provides a systematic approach for set-labeling in graph theory.

Enhances methods for analyzing graph labelings and their properties.

Abstract

Let $X$ denote a set of all non-negative integers and $\sP(X)$ be its power set. A weak integer additive set-labeling (WIASL) of a graph $G$ is an injective set-valued function $f:V(G)\to \sP(X)-\{\emptyset\}$ where induced function $f^+:E(G) \to \sP(X)-\{\emptyset\}$ is defined by $f^+ (uv) = f(u)+ f(v)$ such that either $|f^+ (uv)|=|f(u)|$ or $|f^+ (uv)|=|f(v)|$ , where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. The sparing number of a WIASL-graph $G$ is the minimum required number of edges in $G$ having singleton set-labels. In this paper, we discuss an algorithm for finding the sparing number of arbitrary graphs.