A note on dispensing number algorithm of graphs

Abstract

Let X denote a finite set of non-negative integers and 𝒫(X) be its power set. An integer additive set-labeling (IASL) of a graph G is an injective set-valued function f : V(G) → 𝒫(X) − {∅}, where induced function f⁺ : E(G) → 𝒫(X) − {∅} is defined by f⁺ (uv) = f (u) + f (v), the sumset of f (u) and f (v). An arithmetic integer additive set-labeling (AIASL) of a graph is an IASL under which the set-label of every element of G is an arithmetic progression. A prime AIASL is an AIASL in which the common difference of the set label of any vertex is a prime multiple (or divisor) of the common difference of the set-labels of its adjacent adjacent vertices. The dispensing number of an AIASL-graph G is the minimum number of edges to be removed from G so that it admits a prime AIASL. In this note, we discuss an algorithm for finding the dispensing number of arbitrary graphs.

Keywords:

• Integer additive set-labeled graphs
• arithmetic integer additive set-labeled graphs
• prime arithmetic integer additive set-labeled graphs
• dispensing number of a graph
• dispensing number algorithm

Mathematics Subject Classification:

• 05C78
• 05C85