The Metric Chromatic Number of Zero Divisor Graph of a Ring Z n

Abstract

Let $\mathrm{\Gamma }$ be a nontrivial connected graph, $c:V\left(\mathrm{\Gamma }\right)⟶\mathbb{N}$ be a vertex colouring of $\mathrm{\Gamma }$ , and $L_i$ be the colouring classes that resulted, where $i=\mathrm{1,2},\dots ,k$ . A metric colour code for a vertex a of a graph $\Gamma$ is $c\left(a\right)=\left(d\left(a,L_1\right),d\left(a,L_2\right),\dots ,d\left(a,L_n\right)\right)$ , where $d\left(a,L_i\right)$ is the minimum distance between vertex $a$ and vertex $b$ in $L_i$ . If $c\left(a\right)\ne c\left(b\right)$ , for any adjacent vertices $a$ and $b$ of $\Gamma$ , then $c$ is called a metric colouring of $\Gamma$ as well as the smallest number $k$ satisfies this definition which is said to be the metric chromatic number of a graph $\Gamma$ and symbolized $\mu \left(\mathrm{\Gamma }\right)$ . In this work, we investigated a metric colouring of a graph $\mathrm{\Gamma }\left(Z_n\right)$ and found the metric chromatic number of this graph, where $\mathrm{\Gamma }\left(Z_n\right)$ is the zero-divisor graph of ring $Z_n$ .