graph-theory - Discrete Mathematics Ashwin Nayak U....

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Discrete Mathematics U. Waterloo ECE 103, Spring 2010 Ashwin Nayak June 19, 2010 Some applications of graph theory In these notes, we visit some important applications of elementary Graph Theory. Gray Codes An n -bit Gray code, also called the refected binary code , is an ordering of the 2 n strings of length n over { 0 , 1 } such that every pair of successive strings differ in exactly one position. For example, an 2 -bit Gray code is 00 , 01 , 11 , 10 , and a 3 -bit Gray code is 000 , 001 , 101 , 111 , 011 , 010 , 110 , 100 . If we use successive strings in a Gray code to represent the integers from 0 up to 2 n 1 , instead of the usual binary representation, we see that incrementing a number by one involves ±ipping only one bit. In the usual binary representation, incrementing by 1 could lead to a sequence of bits that are carried over, which may change several consecutive bits at once. For example, 7 + 1 = 8 in the usual binary representation is 111 2 + 1 2 = 1000 2 , and we see that incrementing 111 by 1 causes four bits to be ±ipped. This simple property makes Gray codes very useful in practice. Although they had appeared in mathematical puzzles earlier, Gray codes were proposed in 1947 by Frank Gray, a physicist and researcher at Bell Labs, to prevent spurious output from electromechanical switches. (As you may have guessed, the codes are named after him.) Today, Gray codes are widely used to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems. We have seen Gray codes with 2 and 3 bits. Do they exist for all n ? Below, we see that this question is closely related to a basic property of the n -dimensional hypercube H n . Recall that the vertices of H n are precisely the set of all n -bit strings, and two vertices x,y form an edge iff they differ in exactly in one position. So, the question of existence of an n -bit Gray code may be reformu- lated as: Is there a path in H n that includes all the 2 n vertices? The answer is, as you may have guessed, yes! In fact, the hypercube satis²es a stronger property, that there is a cycle that contains all 2 n of its vertices (when n 2 ). Disregarding any one of the edges in such a cycle gives us the path we seek. Such a cycle is shown in bold for H 2 and H 3 below; H 1 cannot have such a cycle, but has the kind of path we seek. Figure 1: Hamilton paths and cycles in H 1 ,H 2 3 , depicted in bold. DeFnition 1 A cycle in a graph G that contains all the vertices in it is called a Hamilton cycle (sometimes a Hamiltonian cycle). In the last home work, we saw that the grid graph G n has a Hamilton cycle for even n . We now prove that the n -hypercube also does, for all n 2 . In proving this, we make use of the fact that we can construct the ( n + 1) -hypercube by connecting two copies of the n -hypercube with a suitable set of edges. So we can 1
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construct a cycle in the larger hypercube by “cutting and pasting” cycles in the two copies. For example, we obtained the Hamilton cycle in H 3
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graph-theory - Discrete Mathematics Ashwin Nayak U....

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