A9 - t 1 or t 5 . Recompute the bipartite graph (call it H...

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MATH 239 Spring 2010 Assignment 9 Don’t hand in this assignment - do it for ESSENTIAL practice 1. Workers w i are capable of performing jobs t j as shown in the following table. Worker capable of these jobs w 1 t 1 , t 2 , t 3 w 2 t 1 , t 2 , t 5 w 3 t 2 , t 4 , t 6 w 4 t 3 , t 4 , t 5 w 5 t 4 , t 6 w 6 t 2 , t 6 (a) Draw a bipartite graph G to represent this information with vertices the w i and t j and an edge to represent capability. The remaining parts concern the set M containing just the edges { w 1 ,t 1 } , { w 2 ,t 2 } , { w 3 ,t 4 } , { w 4 ,t 5 } , { w 6 ,t 6 } . (b) Is M a matching of G? Why/why not? (c) Is M a perfect matching? Why/why not? (d) Find two alternating paths with respect to M which are not augmenting paths. (e) Find and augmenting path with respect to M , or explain why one does not exist. (f) Give a maximum matching in G . Hence, give an assignment of the workers to the jobs so that the maximum possible number of workers are occupied. (g) Does G have a perfect matching?. (h) Worker w 2 becomes partially incapacitated and can no longer perform tasks
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Unformatted text preview: t 1 or t 5 . Recompute the bipartite graph (call it H now) taking this into account, and then nd a minimum cover and a maximum matching in H , starting with the matching M . Does H have a perfect matching? 2. Let G be a graph with 2 n vertices such that every vertex has degree at least n . Prove that G has a perfect matching. 3. Give an example of a 3-regular graph that does not have a pefect matching. (Note that such a graph cannot be bipartite.) 4. Let G be a bipartite graph with vertex classes A and B , where | A | = | B | = 2 n . Suppose that | N ( X ) | | X | for all subsets X A with | X | n , and | N ( X ) | | X | for all subsets X B with | X | n . Prove that G has a perfect matching....
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