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Unformatted text preview: MATH 239 Assignment 7 This assignment is for practice only, and is not to be handed in. 1. Find a maximum matching and a minimum cover in the graph in Figure 1. Solution: Figure 1: A maximum matching We claim the matching shown in bold in Figure 1 is maximum. To show this, we find a cover of the same size. Following the bipartite matching algorithm in the course notes, let M be the matching above, and V = ( A,B ) with A being the vertices { 1 , 2 ,... , 8 } . Step 1. Set ˆ X = { 7 } , ˆ Y = ∅ . Step 2. Let ˆ Y = { 10 , 12 , 16 } and set pr (10) = pr (12) = pr (16) = 7. Step 3. Step 2 added some vertices to ˆ Y , continue. Step 4. No unsaturated vertices in ˆ Y . Step 5. Add { 1 , 3 , 5 } to ˆ X , and set pr (1) = 10 ,pr (3) = 12 and pr (5) = 16. Now ˆ X = { 1 , 3 , 5 , 7 } . Go to Step 2. Step 2. No new vertices added to ˆ Y . Step 3. M is a maximum matching and the cover C = ˆ Y ∪ ( A \ ˆ X ) = { 2 , 4 , 6 , 8 , 10 , 12 , 16 } is minimum. Indeed the maximum matching above and the minimum cover C = { 2 , 4 , 6 , 8 , 10 , 12 , 16 } have the same size (7). 2. Find a subset D of { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 } such that  N ( D )  < D . Solution: A suitable choice can be found from the proof of Hall’s Theorem as D = ˆ X : so the set D = { 1 , 3 , 5 , 7 } works. Its neighbourhood is { 10 , 12 , 16 } . 3. Let k be a positive integer and suppose G is a bipartite graph in which every vertex has degree precisely k . Show: (a) any bipartition ( A,B ) of G has  A  =  B  Solution:...
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 Spring '09
 M.PEI
 Math, Combinatorics, Bipartite graph, perfect matching

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