In this paper, we study the total outer-connected domination number of the middle graph of a simple graph and we obtain tight bounds for this number in terms of the order of the middle graph. We also compute the total outer-connected domination number of some families of graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the total outer-connected domination number of middle graphs.

A total dominator coloring of a graph [Formula: see text] is a proper coloring of [Formula: see text] in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a total dominator coloring. In this paper, we study the total dominator coloring on middle graphs by giving several bounds for the case of general graphs and trees. Moreover, we calculate explicitly the total dominator chromatic number of the middle graph of several known families of graphs.

No 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 every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a total dominator coloring of it.Here, we study the total dominator coloring on central graphs by giving some tight bounds for the total dominator chromatic number of the central of a graph, join of two graphs and Nordhaus-Gaddum-like relations. Also we will calculate the total dominator chromatic number of the central of a path, a cycle, a wheel, a complete graph and a complete multipartite graph.

The total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. Here, we will find the total dominator chromatic numbers of cycles and paths.