Assessment 6.docx - Assessment 6 1. Let V = {cities of...

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Assessment 61. Let V = {cities of Metro Manila} and E = {(x; y) | x and y areadjacent cities in Metro Manila}.(a) Draw the graph G defined by G = (V; E). You may use initialsto name a vertex representing a city.(b) Apply the Four-Color Theorem to determine the chromaticnumber of the vertex coloring for G.2. Apply Euler’s Theorems and Fleury’s Algorithm to determine Eulerpath and Euler circuits in each graph.Euler’s Theorem 1:• Every vertex of (a) has even degree. Thereforeit has at least one Euler circuit.Euler’s Theorem 2:• The graph has no odd degree, therefore, it hasat least one degree.Euler’s Theorem 3:No of verticesw/ odd degreesNo. of verticeswith even degreesSumNo. ofEdgesa.092814
Fleury’s Algorithm: Path A-B-G-H-C-J-E-H-B-C-D-E-F-G-A is a Eulercircuit.Euler’s Theorem 1:• Two vertices of graph (b) haveodd degree.Therefore it has no Eulercircuit.Euler’s Theorem 2:• Vertices with odd degree is >2. Therefore ithas a Euler path• The graph is connected and has just twovertices of odd degree, therefore it has at leastone Euler pathEuler’s Theorem 3:Fleury’s Algorithm:The path E-F-G-H-L-G-C-H-M-L-K-F-B-EK-J-E-A-B-C-D-H is a Euler circuit.3. A businessman has to visit five cities A, B, C, D and E. The
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Graph Theory, Eulerian path

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