# Understanding some facts about complete graph

Uniquely colorable graph is a graph in which each vertex of chromatic partition has different color.

Does that mean:

1. "Only" complete graphs are uniquely colorable?

Also have following doubts not related to graph coloring:

1. Does the independent set of complete graph contain single vertex?

Your sentence is not well written, it should be

A uniquely colorable graph is a graph in which all sets in a chromatic partition have different colors.

The complete graph is trivially uniquely colorable, but this in not an equivalence. Any tree is uniquely $$2$$-colorable for instance. Even cycles are uniquely $$2$$-colorable too. Odd cycles are not uniquely $$3$$-colorable though.

Does the independent set of complete graph contain single vertex?

Yes. The unique independent sets of a complete graph are made of single vertices.

• The definition of uniquely colorable I mentioned in the question says "each vertex of chromatic partition has different color". Does your definition of "uniquely 2-colorable" also follow this property?
– RajS
Commented Apr 12, 2019 at 18:55
• This sentence does not make any sense. By definition of a chromatic partition, all vertices in the same set of the partition have the same color. This could be a very poorly translation from another anguage maybe. Commented Apr 13, 2019 at 14:43
• hmm...it seems that I ill understood the concept of uniquely colorable graph...In fact, now I feel your correction to my definition of uniquely colorable is insufficient. Wikipedia says: "uniquely colorable graph is a k-chromatic graph that has only one possible k-coloring" and wolfram says: "A uniquely k-colorable graph G is a $\chi$-colorable graph such that every $\chi$-coloring gives the same partition of G".
– RajS
Commented Apr 13, 2019 at 15:23
• These definitions are equivalent to mine. Note that the wikipedia def precises "up to permutation". If the sets on a chromatic partition (which is minimal by definition) are all of different colors, then the coloring is unique (again, up to permutation) Commented Apr 13, 2019 at 16:47