# Total Dominator Chromatic Number of a Hypercube (Solved)

**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 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]). 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.

The following theorems are proved in [Kaz2].

**Theorem**For any integer , .

**Theorem**For any integer , .

**Theorem**1. If , then .

2. If , then .

3. If , then .

## 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.

## False

Since it follows that for large enough the conjecture does not hold by the same token as written in the comment here http://www.openproblemgarden.org/op/total_domination_number_of_a_hypercube