By Alan Tucker

ISBN-10: 0470458380

ISBN-13: 9780470458389

This can be Alan Tuckers textbook on combinatorics and graph conception

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Extra info for Applied Combinatorics (6th Edition)

Example text

At any given vertex, there will be edges going to each of the n − 1 other vertices in K n , and so each vertex has degree n − 1. The sum of the degrees of all n vertices in K n will be n(n − 1). Since this sum equals twice the number of edges, the number of edges is n(n − 1)/2. Example 3: Impossible Graph Is it possible to have a group of seven people such that each person knows exactly three other people in the group? If we model this problem using a graph with a vertex for each person and an edge between each pair of people who know each other, then we would have a graph with seven vertices all of degree 3.

A) Show that every circuit in the graph in Exercise 3(l) has at least five edges. (b) Use part (a) and the result of Exercise 20 to show that this graph is nonplanar. 22. (a) Give an example of a graph with regions consisting solely of squares (regions bounded by four edges) and hexagons, and with vertices of degree at least 3.

By rotational symmetry, we can match a to any vertex in the right graph (that is, if the two graphs are isomorphic, there will exist an isomorphism with a matched to any vertex in the right graph). Let us use the match a − 1. The set of neighbors of a (vertices adjacent to a) must be matched with the set of neighbors of 1. Let us look at the subgraphs formed by these neighbors of a and 1. 9. Both subgraphs are paths: one is f to g to b to c, and the other is 7 to 4 to 5 to 2. The isomorphism must make these path subgraphs isomorphic.

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Applied Combinatorics (6th Edition) by Alan Tucker


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