دوستعلی مژده

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.