Unformatted text preview: Math 239
Assignment 11
Not to be turned in. Solutions will be posted Monday July 26th.
Covering sections 7.27.4. 1. Show that X  = X0  + Y  − U  U where is the set of unsaturated vertices in the XY construction.
2. Let G A, B , be a bipartite graph with bipartition (a) Show that, for any M and be a matching of G. D ⊆ A, M  ≤ A − D + N (D). (b) Show that the size of a maximum matching in G is equal to min A − D + N (D) . D ⊆A 3. Use Hall's theorem to prove that the following bipartite graph G does not have a perfect matching. 1 2 k regular 9 5 be a 8 4 G 7 3 4. Let 6 10 bipartite graph. (a) Use Theorem 7.3.4 and induction on k to show that G has k perfect matchings that are disjoint (i.e., any two have no edge in common).
(b) Use part (a) to show that
(c) A 01 G is edge k colourable. matrix has entries each being either permutation matrix if there is exactly one
Suppose N is an n × n 01 use part (b) to prove that 5. 0
1 matrix with exactly N is the sum of (a) Show that Petersen's graph is not edge k or 1. An n × n 01 matrix is a in each row and in each column. k 1's in each row and each column, permutation matrices. 3colourable. (b) Why does part (a) not violate the conclusion of question 4(b)?
(c) Show that Petersen's graph is edge
6. We say that a component H 4colourable. of a graph is odd if V (H ) is odd. S ⊆ V (G). Let G − S denote the subgraph G of G induced by
deleting the vertices in S and any edge incident to some vertex in S . Show that if the
number of odd components in G − S is greater than S , then G does not admit a perfect
Let G be a graph and matching. ...
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 Spring '09
 M.PEI
 Math, Bipartite graph, 1 g, Petersen, 1 2 K, 8 4 G

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