How can we prove that a coin graph is 4-colorable???Also, can we find any example of an non-3-colorable coin graph.



Coin graph is 4-colorable
The phrase "coin graph" sometimes requires equal coins and sometimes allows non-equal coins. I am assuming the former meaning as indicated in the comments: equal coins.


[This paper](http://infoscience.epfl.ch/record/129193/files/coincikk.ps) says that the 4-colourability can be proved using a "simple induction". I'll try: Consider the convex hull of the coin centres. A coin whose centre is a vertex of the convex hull is adjacent to at most 3 other coins, so you can always colour it after you colour the others.


Finally, the example shown in Fig 1 of that paper is not 3-colourable.


