By Herbert S. Wilf

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**Extra info for Algorithms and Complexity (Internet edition, 1994)**

**Example text**

The complete bipartite graph Km,n has a set S of m vertices and a set T of n vertices. Its edge set is E(Km,n ) = S × T . It has |E(Km,n )| = mn edges. The n-cycle, Cn , is a graph of n vertices that are connected to form a single cycle. A tree is a graph that (a) is connected and (b ) has no cycles. A tree is shown in Fig. 6. Fig. 6: A tree 27 Chapter 1: Mathematical Preliminaries It is not hard to prove that the following are equivalent descriptions of a tree. (a) A tree is a graph that is connected and has no cycles.

First, just for general cultural reasons, let’s slow down for a while and discuss the relationship between graph colorings in general and the four-color problem, even though it isn’t directly relevant to what we’re doing. The original question was this. Suppose that a delegation of Earthlings were to visit a distant planet and find there a society of human beings. , and of course, all at war with each other. The delegation wants to escape quickly, but before doing so it draws a careful map of the 5,000,000 countries into which the planet has been divided.

Research problem) Find the asymptotic behavior, for large n, of the probability that a randomly chosen permutation of n letters has a splitter. 3 Recursive graph algorithms Algorithms on graphs are another rich area of applications of recursive thinking. Some of the problems are quite different from the ones that we have so far been studying in that they seem to need exponential amounts of computing time, rather than the polynomial times that were required for sorting problems. We will illustrate the dramatically increased complexity with a recursive algorithm for the ‘maximum independent set problem,’ one which has received a great deal of attention in recent years.

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

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