Lecture3 - CME 305: Discrete Mathematics and Algorithms...

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CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi (saberi@stanford.edu) January 12, 2009 Lecture 3: Eulerian Circuits This lecture is about one of the first known applications of graph theory, Eulerian circuits. We give some basic definitions, prove Euler’s result, and then illustrate an application from bio-informatics. 1 Eulerian Circuits Definition: A closed walk (circuit) on graph G ( V,E ) is an Eulerian circuit if it traverses each edge in E exactly once. We call a graph Eulerian if it has an Eulerian circuit. The problem of finding Eulerian circuits is perhaps the oldest problem in graph theory. It was originated by Euler in the 18th century; the problem was whether one could take a walk in K¨onigsberg and cross each of the four bridges exactly once. Motivated by this, Euler was able to prove the following theorem: Theorem 1 (Euler, 1736) Graph G ( V,E ) is Eulerian iff G is connected (except for possible isolated vertices) and the degree of every vertex in G is even. Proof: ”: assume G has an Eulerian circuit. Clearly, G must be connected, otherwise we will be unable to traverse all the edges in a closed walk. Suppose that an Eulerian circuit starts at v 1 ; note that every time the walk enters vertex v 6 = v 1 , it must leave it by traversing a new edge. Hence, every time that the walk visits v it traverses two edges of v . Thus d v is even. The same is true for v 1 except for the first step of the walk that leaves v 1 and the last step that it enters v 1 . Again, we can pair the first and last edges in the circuit to conclude that d v 1 is also even. ”: we prove this by strong induction on the number of nodes. For n = 1, the statement is trivial. Assume G has k + 1 vertices, it is connected, and all the nodes have even degrees. Pick any vertex v V and start walking through the edges of the graph, traversing each edge at most once. Given that all the degrees of the nodes in G are even, we can only get stuck at v . Remove the circuit of visited edges, c 1 , from G ; Call this new graph G 0 . Node v is an isolated vertex in
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Lecture3 - CME 305: Discrete Mathematics and Algorithms...

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