(Challenging problem) The five-color theorem states that the chromatic number for any simple,…

(Challenging problem) The five-color theorem states that the chromatic number for any simple, connected, planar graph is at most 5. While the four-color theorem (Exercise 72) is very difficult to prove, the five-color theorem can be proved by induction on the number of nodes in the graph. Prove the five-color theorem, making use of the result in Exercise 73.

Exercise 72

At least four colors are required to solve the general map-coloring problem (see Exercise 69). Because no one could produce a map requiring more than four colors, the conjecture was formulated that four colors are indeed sufficient. This conjecture became known as the four-color problem. It was first proposed to the mathematician Augustus De Morgan by one of his students in 1852, and it subsequently received much attention. It remained unproved, however, for over 100 years. In 1976 two mathematicians at the University of Illinois, Wolfgang Haken and Kenneth Appel, used a computer to work though a large number of cases in a proof by contradiction, thus verifying the four-color conjecture.

                    The dual graph for a map (see Exercise 70), by the way it is constructed, will always be simple, connected, and planar. In addition, any simple, connected, planar graph can be viewed as the dual graph of a map. Restate the four-color conjecture in terms of the chromatic number (see Exercise 71) of a graph

Exercise 69

Draw a map that requires four colors.

Exercise 71

A coloring of a graph is an assignment of a color to each node of the graph in such a way that no two adjacent nodes have the same color. The chromatic number of a graph is the smallest number of colors needed to achieve a coloring. Find the chromatic number of the following graphs.

(Challenging problem) The five-color theorem states that the chromatic number for any simple,...

Exercise 73

Prove that in a simple, connected, planar graph with three or more nodes, there is at least one node with degree less than or equal to 5. (Hint: Use a proof by contradiction.)

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