**Conjecture**For any integer , .

Here denotes the -dimensional hypercube, i.e. the graph with vertex set and two vertices adjacent if they differ in exactly one coordinate. A *total dominating set* of a graph is a set of vertices of such that every vertex has at least one neighbor in ". The *total domination number* of is the cardinality of a minimum total dominating set. A *total dominator coloring* of a graph , briefly TDC, is a proper coloring of in which each vertex of the graph is adjacent to every vertex of some color class. The *total dominator chromatic number* of is the minimum number of color classes in a TDC in (see [Kaz1]).

The following theorems are proved in [Kaz2].

**Theorem**For any integer , .

**Theorem**1. If , then .

2. If , then .

3. If , then .

**Theorem**For any integer , .

## Bibliography

[Kaz1] Adel P. Kazemi, Total dominator chromatic number of a graph, http://arxiv.org/abs/1307.7486.

[Kaz2] Adel P. Kazemi, Total Dominator Coloring in Product Graphs, Utilitas Mathematica (2013), Accepted.

* indicates original appearance(s) of problem.