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MIT6_042JF10_assn04 - 6.042/18.062J Mathematics for...

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6.042/18.062J Mathematics for Computer Science September 28, 2010 Tom Leighton and Marten van Dijk Problem Set 4 Problem 1. [15 points] Let G = ( V, E ) be a graph. A matching in G is a set M E such that no two edges in M are incident on a common vertex. Let M 1 , M 2 be two matchings of G . Consider the new graph G = ( V, M 1 M 2 ) (i.e. on the same vertex set, whose edges consist of all the edges that appear in either M 1 or M 2 ). Show that G is bipartite. Helpful definition : A connected component is a subgraph of a graph consisting of some vertex and every node and edge that is connected to that vertex. Problem 2. [20 points] Let G = ( V, E ) be a graph. Recall that the degree of a vertex v V , denoted d v , is the number of vertices w such that there is an edge between v and w . (a) [10 pts] Prove that 2 | E | = d v . v V (b) [5 pts] At a 6.042 ice cream study session (where the ice cream is plentiful and it helps you study too) 111 students showed up. During the session, some students shook hands with each other (everybody being happy and content with the ice-cream and all). Turns out that the University of Chicago did another spectacular study here, and counted that each student shook hands with exactly 17 other students. Can you debunk this too? (c) [5 pts] And on a more dull note, how many edges does K n , the complete graph on n vertices, have?
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