TITLE: Colored graph with $2$ colors
QUESTION [6 upvotes]: Let us call $G$ a graph with vertices in two possible colours. If we select a vertex, we change the colour of it and of every vertex that is adjacent to it. Is it possible to change a graph from all the first colour to all the second using such moves?

I cannot find any counter example but either cannot find a proof. What do you think?

REPLY [4 votes]: Yes, it can be done for any graph. This is a much-discussed problem called the "lamp lighter's problem" or the "all ones problem" or "lights out". This paper which is available online looks like a good survey:
Rudolf Fleischer and Jiajin Yu, A Survey of the Game "Lights Out".