Wikipedia showcases a penny graph that requires 4 colors for a proper coloring on the vertices. Why does this graph require 4 colors instead of 3?
More Generally I'm looking for a general condition in a penny graph $G$ that results in $\chi(G)=4$.
I've been told that looking at the vertices farthest from each other should help, but I don't know what about them to consider. In wikipedia's example, all I see is that these vertices each have degree 3, and their vertex induced subgraphs will have coloring in 3 colors. Then I guess you could argue that the rest of the vertices have uniquely defined colors that lead to a necessary 4th color, but how would I formalize and/or generalize this?