American Mathematical Monthly, volume 117, number 1, january by Daniel J. Velleman PDF

By Daniel J. Velleman

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Extra info for American Mathematical Monthly, volume 117, number 1, january 2010

Example text

5. (k, m)-AGREEABLE LINEAR SOCIETIES. We now use the connection between perfect graphs, the clique number, and the chromatic number to obtain a lower bound for the agreement number of a (k, m)-agreeable linear society (Theorem 8). We first need a lemma that says that in the corresponding agreement graph, the (k, m)agreeable condition prevents any coloring of the graph from having too many vertices of the same color. Thus, there must be many colors and, since the graph is perfect, the clique number must be large as well.

The perfect graph property will allow us, in the next section, to make a crucial connection between the (k, m)-agreeability condition and the agreement number of the society. Given its importance in our setting, it is worth making a few comments about how perfect graphs appear in other contexts in mathematics, theoretical computer science, and operations research. The concept was introduced in 1961 by Berge [2], who was motivated by a question in communication theory, specifically, the determination of the Shannon capacity of a graph [19].

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American Mathematical Monthly, volume 117, number 1, january 2010 by Daniel J. Velleman


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