By Herbert S. Wilf
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Extra resources for Algorithms and Complexity (Internet edition, 1994)
It was then conjectured that four colors are always sufficient for the proper coloring of the countries of any map at all. 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.
To construct a graph we would decide which of these possible edges would be used. We can make each of these n2 decisions independently, and for every way of deciding where to put the edges we would get a different graph. Therefore the number of n labeled graphs of n vertices is 2( 2 ) = 2n(n−1)/2. If we were to ask the corresponding question for unlabeled graphs we would find it to be very hard. The answer is known, but the derivation involves Burnside’s lemma about the action of a group on a set, and some fairly delicate counting arguments.
How many bits of input data does it take to describe a graph? Well, certainly we can march through the entire list of n(n − 1)/2 pairs of vertices and check off the ones that are actually edges in the input graph to the problem. Hence we can describe a graph to a computer by making 39 Chapter 2: Recursive Algorithms Fig. 3: A tree-full of graphs is created a list of n(n − 1)/2 0’s and 1’s. Each 1 represents a pair that is an edge, each 0 represents one that isn’t an edge. Thus Θ(n2 ) bits describe a graph.
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