Graph Theory

Discrete groups, expanding graphs and invariant measures by Lubotzky A.

By Lubotzky A.

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Extra resources for Discrete groups, expanding graphs and invariant measures

Example text

An algorithm for 11 is a step-by-step procedure which when applied to any instance of 11 produces a solution. Usually we can rewrite an optimization problem as a decision problem which at first seems to be much easier to solve than the original but turns out to be just about as hard. Consider the following two versions of the graph coloring problem. GRAPH COLORING (optimization version) Instance: An undirected graph G. Question: What is the smallest number of colors needed for a proper coloring ofG?

Vi, VQ] is called a cycle of length / -f 1 (or closed path) if Vi^ iVi e £ for / = 1, 2 , . . , / and VIVQ e E. Simple cycle: A cycle [VQ, Vi, V2,. , Vi, VQ] is a, simple cycle if Vi ^ Vj for Chordless cycle: A simple cycle [vo,Vi,V2,^",Vi,Vo]is for i a n d ; differing by more than 1 mod / + 1. , every edge has one endpoint in Si and the other in S2. Equivalently, G is bipartite if and only if it is 2-colorable. It is customary to use the notation G = (Si, S2, E), which emphasizes the partition.

4. Summary The reader has been introduced to the graph theoretic foundations needed for the remainder of the book. In addition, he has had a taste of some of the particular notions that we intend to investigate further. Returning to the table of contents at this point, he will recognize many of the topics listed. 15. 15. The chapter dependencies. The reader may wish to read Chapters 1 and 2 quickly and refer back to them as needed. 18 1. Graph Theoretic Foundations In the next chapter we will present the foundations of algorithmic design and analysis.

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