Homework 3-Solutions copy

Homework 3-Solutions copy - HOMEWORK 3 SOLUTIONS (1) Show...

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Unformatted text preview: HOMEWORK 3 SOLUTIONS (1) Show that for each n N the complete graph K n is a contraction of K n,n . Solution: We describe the process for several small values of n . In this way, we can discern the inductive step. Clearly, K 1 , which is just one vertex, is a simple contraction of K 1 , 1 , which is simply one edge along with its endvertices. e 1 K 1 , 1 K 1 Now consider the graph K 2 , 2 , represented in the usual way. A simple contraction on the right vertical edge followed by a simple contraction on the left vertical edge produces a graph that consists of a single edge, i.e., K 2 . e 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 e 1 e 2 H H H H H H H H H H H H H H H H H H H H H H H K 2 , 2 K 2 y y y y y y y y y y y y y y y y y y y y y y y y 2 SOLUTIONS We continue on to K 3 , 3 . We perform simple contractions on the vertical edges in sequence, starting from the right. e 3 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H e 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 e 1 v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v K 3 , 3 e 3 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U e 2 J J J J J J J J J J J J J J J J J J J J J J J i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i t t t t t t t t t t t t t t t t t t t t t t t e 3 J J J J J J J J J J J J J J J J J J J J J J J U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i t t t t t t t t t t t t t t t t t t t t t t t HOMEWORK 3 3 K 3 k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k x x x x x x x x x x x x x x x x x x x x x x x x The pattern has now becomes a little bit clearer. When we performed a simple contraction on the edge e 1 of K 3 , 3 the result was a graph with a single vertex, a K 1 , on the right connected to every vertex of a K 2 , 2 on the left. After performing a simple contraction on the edge e 2 , the resulting graph had two adjacent vertices on the right, a K 2 , that were each connected to every vertex of a K 1 , 1 on the left. The last simple contraction, on edge e 3 , produced a graph on three vertices, where every pair of vertices are adjacent. This, of course, is the definition of K 3 ....
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Homework 3-Solutions copy - HOMEWORK 3 SOLUTIONS (1) Show...

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