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# pset6 - EECS 310 Fall 2011 Instructor Nicole Immorlica...

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Unformatted text preview: EECS 310, Fall 2011 Instructor: Nicole Immorlica Problem Set #6 Due: November 8, 2011 1. (25 points) Consider the following false claim: Let G be a simple graph with maximum degree at most k . If G has a vertex of degree less than k , then G is k-colorable. (a) (5 points) Give a counterexample to the false claim when k = 2. (b) (15 points) Consider the following proof of the false claim: Proof. Proof by induction on the number of vertices n . • Inductive Hypothesis : P ( n ) is defined to be: Let G be a graph with n vertices and maximum degree at most k . If G has a vertex of degree less than k , then G is k-colorable. • Base Case : ( n = 1) G has only one vertex and is 1-colorable, so P (1) holds. • Inductive Step : We may assume P ( n ). To prove P ( n +1), let G n +1 be a graph with n + 1 vertices and maximum degree at most k . Also, suppose G n +1 has a vertex v of degree less than k . We need only prove that G n +1 is k-colorable....
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pset6 - EECS 310 Fall 2011 Instructor Nicole Immorlica...

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