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Unformatted text preview: vertices. (Vertex v j precedes v i if j < i .) Use induction to prove that every graph with width at most w is ( w + 1)colorable. (Recall that a graph is kcolorable iﬀ every vertex can be assigned one of k colors so that adjacent vertices get diﬀerent colors.) 3 Problem 2 A planar graph is a graph that can be drawn without any edges crossing. 1. First, show that any subgraph of a planar graph is planar. 2. Also, any planar graph has a node of degree at most 5. Now, prove by induction that any graph can be colored in at most 6 colors. MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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 Fall '10
 TomLeighton,Dr.MartenvanDijk
 Computer Science

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