The Moore graphs; total domination and total dominator chromatic numbers

Abstract

‎A total dominator coloring of a graph $G$ is a proper coloring of $G$ in which each vertex of the graph is adjacent to all vertices of a color class‎. ‎The minimum number of color classes in a total dominator coloring of a graph is called its total dominator chromatic number‎. ‎Here‎, ‎we study the total domination and total dominator chromatic numbers of Moore graphs which are a family of $k$-regular graph of girth $g$ and have the smallest order $n_0(k,g)$ (a known number in terms of $k$ and $g$)‎.