By Herbert S. Wilf

ISBN-10: 1568811780

ISBN-13: 9781568811789

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**Extra resources for Algorithms and Complexity**

**Sample text**

We speak of Hamiltonian and Eulerian circuits of G as circuits of G that visit, respectively, every vertex, or every edge, of a graph G. There is a world of diﬀerence between Eulerian and Hamiltonian paths, however. If a graph G is given, then thanks to the following elegant theorem of Euler, it is quite easy to decide whether or not G has an Eulerian path. In fact, the theorem applies also to multigraphs, which are graphs except that they are allowed to have several diﬀerent edges joining the same pair of vertices.

Since a0 and c0 are both equal to n mod b, they are equal to each other. Hence the number n0 = (n − a0 )/b has two diﬀerent representations, which contradicts the inductive assumption, since we have assumed the truth of the result for all n0 < n. The bases b that are the most widely used are, aside from 10, 2 (‘binary system’), 8 (‘octal system’) and 16 (‘hexadecimal system’). 22 1. Mathematical Preliminaries The binary system is extremely simple because it uses only two digits. This is very convenient if you’re a computer or a computer designer, because the digits can be determined by some component being either ‘on’ (digit 1) or ‘oﬀ’ (digit 0).

Suppose first¡ that we mean labeled graphs. A graph of n vertices has ¢ a maximum of n2 edges. 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 labeled n 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.

### Algorithms and Complexity by Herbert S. Wilf

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