Total gamma coloring in graphs

A coloring of a graph [Formula: see text] is the assignment of colors to the vertices of [Formula: see text] such that no two adjacent vertices receive the same color. The minimum number of colors required to color [Formula: see text] is said to be the chromatic number of [Formula: see text], and is denoted by [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is said to be a dominating set of [Formula: see text] if every vertex in [Formula: see text] is adjacent to some vertex in [Formula: see text]. If the graph induced on [Formula: see text] has no isolates, the [Formula: see text] is said to be a total dominating set of [Formula: see text]. The minimum cardinality of a dominating (total dominating) set of [Formula: see text] is said to be the domination (total domination) number of [Formula: see text], and is denoted by [Formula: see text] ([Formula: see text]). For a graph [Formula: see text] without isolated vertices, a total gamma coloring is a coloring of [Formula: see text], if there is a total dominating set [Formula: see text] of [Formula: see text] such that all vertices in [Formula: see text] receive distinct colors. The minimum number of colors required for a total gamma coloring of [Formula: see text] is called the total gamma chromatic number, and is denoted by [Formula: see text]. In this paper, we investigate [Formula: see text] number of graphs, bounds in terms of chromatic number and total domination number, and characterization of extremal graphs attaining a given [Formula: see text] number.