June 10, 2023

# Doost Ali Mojdeh

Degree: Professor
Address: Department of Mathematics, University of Mazandaran, Babolsar, Iran
Education: Ph.D in Mathematics (Graph Theory and Combinatorics)
Phone: 011-35302448
Faculty: Faculty of Mathematical Sciences

## Research

 Title TOTAL GLOBAL DOMINATOR CHROMATIC NUMBER OF GRAPHS Type Article Keywords Total global domination, coloring, dominator coloring, total global dominator coloring Journal TWMS Journal of Applied and Engineering Mathematics DOI Does not have doi Researchers a { color: #4f98b0; } a:hover { color: #ffab00; } a:link:visited { text-decoration: none; } Saeid Askari (First researcher) , Doost Ali Mojdeh (Second researcher) , Esmail Nazari (Third researcher)

## Abstract

Let $G=(V,E)$ be $k$-colorable ($k$-vertex colorable) graph and $V_i\subseteq V$ be the class of vertices with color $i$. Then we assume that $f = (V_{1}, V_{2}, \cdots, V_{k})$ is a coloring of $G$. A vertex $v \in V (G)$ is a dominator of $f$ if $v$ dominates all the vertices of at least one color class such as $V_{i}$ ( $V_i$ is called a dom-color class respected to $v$) and $v$ is said to be an anti dominator of $f$ if $v$ dominates none of the vertices of at least one color class such as $V_{i}$ ( $V_i$ is called a anti dom-color class respected to $v$). A vertex $v \in V (G)$ is a total dominator of $f$, if $v$ dominates all the vertices of at least one color class such as $V_{i}$ not including $v$ ($V_i$ is called a total dom-color class respected to $v$). A total global dominator coloring of a graph $G$ is a proper coloring $f$ of $G$ in which each vertex of the graph has a total dom-color class and an anti dom-color class in $f$. The minimum number of colors required for a total global dominator coloring of $G$ is called the total global dominator chromatic number and is denoted by $\chi^{t}_{gd}( G)$. In this paper we initiates a study on this notion of total global dominator coloring. The complexity of total global dominator coloring is studied. Some basic results and some bounds in terms of order, chromatic number, domination parameters are investigated. Finally we classify the total global dominator coloring of trees.