By Gary Chartrand, Ping Zhang

ISBN-10: 1584888008

ISBN-13: 9781584888000

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**Sample text**

Figure 14: An unavoidable set of configurations in a cubic map A region surrounded by k neighboring regions is called a k-gon. It is possible to show that any map that contains no k-gon where k < 5 must contain at least 18 CHAPTER 0. THE ORIGIN OF GRAPH COLORINGS twelve pentagons (5-gons). In fact, there is a map containing exactly twelve regions, each of which is a pentagon. Such a map is shown in Figure 15, where one of the regions is the “exterior region”. Since this map can be colored with four colors, any counterexample to the Four Color Conjecture must contain at least thirteen regions.

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### Chromatic Graph Theory at BiggerBooks.com by Gary Chartrand, Ping Zhang

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