On Integer Additive Set-Valuations of Finite Jaco Graphs

N. K. Sudev, K. P. Chithra, K. A. Germina

Abstract


Let \(X\) denote a set of non-negative integers and \(\mathcal{P}(X)\) be its power set. An integer additive set-labeling (IASL) of a graph \(G\) is an injective set-valued function \(f:V(G)\to \mathcal{P}(X)-\{\emptyset\}\) where induced function \(f^+:E(G) \to \mathcal{P}(X)-\{\emptyset\}\) is defined by \(f^+ (uv) = f(u)+ f(v)\), where \(f(u)+f(v)\) is the sumset of \(f(u)\) and \(f(v)\). Let \(f(x)=mx+c\); \(m\in \mathbb{N}\), \(c\in N_0\). A finite linear Jaco graph, denoted by \(J_n(f(x))\), is a directed graph with vertex set \(\{v_i: i\in \mathbb{N}\}\) such that \((v_i,v_j)\) is an arc of \(J_n(f(x))\) if and only if \(f(i)+i-d^-(v_j)\ge j\). In this paper, we discuss the admissibility of different types of integer additive set-labeling by finite linear Jaco graphs.

Keywords


Integer additive set-labeled graphs; Weak integer additive set-labeled graphs; Arithmetic integer additive set-labeled graphs; Dispensing number of a graph; Finite linear Jaco graph

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DOI: http://dx.doi.org/10.26713%2Fjims.v8i2.408

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