MIT6_042JF10_rec06 - vertices(Vertex v j precedes v i if...

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6.042/18.062J Mathematics for Computer Science September 29, 2010 Tom Leighton and Marten van Dijk Problems for Recitation 6 1 Graph Basics Let G = ( V,E ) be a graph. Here is a picture of a graph. A B C D F E G Recall that the elements of V are called vertices, and those of E are called edges. In this example the vertices are { A,B,C,D,E,F,G } and the edges are { A B,B D,C D,A C,E F,C E,E G } . Deleting some vertices or edges from a graph leaves a subgraph . Formally, a subgraph of G = ( V,E ) is a graph G = ( V ,E ) where V is a nonempty subset of V and E is a subset of E . Since a subgraph is itself a graph, the endpoints of every edge in E must be vertices in V . For example, V = { A,B,C,D } and E = { A B,B D,C D,A C } forms a subgraph of G . In the special case where we only remove edges incident to removed nodes, we say that G is the subgraph induced on V if E = { ( x y | x,y V and x y E } . In other words, we keep all edges unless they are incident to a node not in V . For instance, for a new set of vertices V = { A,B,C,D } , the induced subgraph G has the set of edges E = { A B,B D,C D,A C } . 2 Problem 1 An undirected graph G has width w if the vertices can be arranged in a sequence v 1 , v 2 , v 3 , ..., v n
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Recitation 6 2 such that each vertex v i is joined by an edge to at most w preceding
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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 k-colorable 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 6.042J / 18.062J Mathematics for Computer Science Fall 2010 For information about citing these materials or our Terms of Use, visit: ....
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MIT6_042JF10_rec06 - vertices(Vertex v j precedes v i if...

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