> a map the requires four colors has four nodes and 6 arcs That's not the only case - there are an infinite number graphs that require 4 colors. It's possible

> a map the requires four colors has four nodes and 6 arcs That's not the only case - there are an infinite number graphs that require 4 colors. It's possible that all of them have a subgraph with four nodes and 6 edges connecting them, but I don't know either way. > So a map that would require five colors should have five nodes and ten arcs How do you know there isn't a graph with 35 nodes and 127 edges that requires 5 colors?