mgf1107lecture14

mgf1107lecture14 - Ex. Recall that we can represent the...

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MGF 1107 – EXPLORATIONS IN MATHEMATICS LECTURE 14 We are now in a position to introduce Euler’s theorem, which is used to determine if a graph contains Euler paths and Euler circuits. Euler’s Circuit Theorem i) If a graph is connected and every vertex is even, then there exists at least one Euler circuit (which we can start and end at any vertex). ii) If any graph has an odd vertex then no Euler circuits exist. Ex.
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Euler’s Path Theorem i) If a graph is connected and has exactly two odd vertices, then it has at least one Euler path. Any such path must start at one of the odd vertices and end at the other. ii) If any graph has more than two odd vertices then it cannot have an Euler path. Ex. Ex.
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We now use Euler’s theorem to solve the Königsberg bridge problem.
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Unformatted text preview: Ex. Recall that we can represent the Knigsberg bridge problem using the graph below. Ex. Ex. If we carefully read through the two theorems described earlier, we see that they do not consider the case where exactly one odd vertex appears in a graph. The following theorem explains that this situation is impossible. Eulers Sum of Degrees Theorem i) The sum of the degrees of all the vertices of a graph equals twice the number of edges (and is hence even). ii) A graph always has an even number of odd vertices. So to summarize, if you calculate the number of odd vertices, then the following conclusions can be reached. Number of Odd Vertices Conclusion...
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mgf1107lecture14 - Ex. Recall that we can represent the...

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