The Metric Chromatic Number of Zero Divisor Graph of a Ring ${\mathrm{Z}}_{\mathrm{n}}$

Abstract

Let Γ be a nontrivial connected graph, c : V Γ ⟶ ℕ be a vertex colouring of Γ , and L i be the colouring classes that resulted, where i = 1,2 , … , k . A metric colour code for a vertex a of a graph Γ is c a = d a , L 1 , d a , L 2 , … , d a , L n , where d a , L i is the minimum distance between vertex a and vertex b in L i . If c a ≠ c b , for any adjacent vertices a and b of Γ , then c is called a metric colouring of Γ as well as the smallest number k satisfies this definition which is said to be the metric chromatic number of a graph Γ and symbolized μ Γ . In this work, we investigated a metric colouring of a graph Γ Z n and found the metric chromatic number of this graph, where Γ Z n is the zero-divisor graph of ring Z n .