# Does a k-colourable graph need to be connected?

This is probably a silly question but I have a definition in front of me that says:

A graph G is k-colourable if the nodes of G can be coloured using no
more than k colours.


Does a colourable graph need to be connected? I think it does as we could have an arbitrary amount of singleton nodes.

• The definition you recall seems somewhat incomplete as you do not mention any conditio on the clorouring. (Perhaps this is implict in the "coloured" in the source you quote.) – quid Feb 20 '15 at 14:41

It is not necessary it is connected. However, a graph is $k$-colourable if and only if each of its connected components is. Thus, it does not change that much.