# Total Dominator Chromatic Number of a Hypercube (Solved)

 Importance: Medium ✭✭
 Subject: Graph Theory » Coloring » » Vertex coloring
 Keywords: Total dominator chromatic number, hypercube.
 Posted by: Adel P. Kazemi on: October 6th, 2013
 Solved by: See comment.
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