Total dominator total chromatic numbers of cycles and paths

Abstract

The total dominator total coloring of a graph is a total coloring of the graph such that each object (vertex or edge) of the graph is adjacent or incident to every object of some color class. The minimum number of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. In \cite{KKM2019}, the authors initiated to study the total dominator total coloring of a graph and found some useful results, and presented some problems. Finding the total dominator total chromatic numbers of cycles and paths were two of them which we consider them here.

Total dominator total coloring of a graph

“…Since the problem of finding total dominator chromatic number of a graph is NP-complete [14], the problem of finding total dominator total chromatic number of a graph is also NP-complete by Theorem 1.4. Also since for any graph G with no isolated vertex of order n and size m, γ t (G) ≤ χ t d (G) ≤ n, by Theorem 1.5, and every TDTC is a total coloring, and also…”

“…Motivated by the relation between coloring and total dominating, the concept of total dominator coloring in graphs introduced in [14] by Kazemi, and extended in [12,13,15,16,17]. The reader can study section 4 of Part I from the book [10] for more information on this concept.…”

“…The reader can study section 4 of Part I from the book [10] for more information on this concept. Definition 1.2 (Kazemi, [14]). A total dominator coloring, briefly TDC, of a graph G with a positive minimum degree is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some color class.…”

Domination Coloring of Graphs

“…More results on the dominator coloring can be found in [23][24][25][26]. Kazemi [27] proposed the concept of total dominator coloring in 2015, which is a proper coloring, such that each vertex of the graph is adjacent to every vertex of some (other) color class. For more results on the total dominator coloring, refer to [28][29][30].…”

Total Dominator Colorings of P 4 -reducible and P 4 -tidy Graphs

“…For convenience, we abbreviated write TD-coloring for total dominator coloring. The total dominator chromatic number χ t d (G) of G is the minimum number of color classes in a T D-coloring of G. It was introduced by Kazemi in [2] and studied further, the more details can refer to [3,4].…”