The key observation is that while it is true that for any number \(n\text{,}\) there is a graph with chromatic number \(n\text{,}\) only some graphs arrive as representations of maps. If you convert a map to a graph, the edges between vertices correspond to borders between the countries. So you should be able to connect vertices in such a way where the edges do not cross. In other words, the graphs representing maps are all planar!
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