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By Herbert S. Wilf

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6 Graphs the two endpoints of e are different. Fig. 4(a) shows a graph G and an attempt to color its vertices properly in 3 colors (‘R,’ ‘Y’ and ‘B’). The attempt failed because one of the edges of G has had the same color assigned to both of its endpoints. In Fig. 4(b) we show the same graph with a successful proper coloring of its vertices in 4 colors. Fig. 4(a) Fig. 4(b) The chromatic number χ(G) of a graph G is the minimum number of colors that can be used in a proper coloring of the vertices of G.

Begin at the same v and walk along 0 or more edges of W until you arrive for the first time at a vertex q of component C1 . This will certainly happen because G is connected. Then follow the Euler tour of the edges of C1 , which will return you to vertex q. , and the proof is complete. It is extremely difficult computationally to decide if a given graph has a Hamilton path or circuit. We will see in Chapter 5 that this question is typical of a breed of problems that are the main subject of that chapter, and are perhaps the most (in-)famous unsolved problems in theoretical computer science.

Settling this conjecture turned out to be a very hard problem. It was finally solved in 1976 by K. Appel and W. Haken* by means of an extraordinary proof with two main ingredients. First they showed how to reduce the general problem to only a finite number of cases, by a mathematical argument. Then, since the ‘finite number’ was over 1800, they settled all of those cases with quite a lengthy computer calculation. So now we have the ‘Four Color Theorem,’ which asserts that no matter how we carve up the plane or the sphere into countries, we will always be able to color those countries with at most four colors so that countries with a common frontier are colored differently.

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Algorithms and Complexity (Internet edition, 1994) by Herbert S. Wilf

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