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Unformatted text preview: Math 239
Not to be turned in. Solutions will be posted Monday July 26th.
Covering sections 7.2-7.4. 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 0-1 G is edge k -colourable. matrix has entries each being either permutation matrix if there is exactly one
Suppose N is an n × n 0-1 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 0-1 matrix is a in each row and in each column. k 1's in each row and each column, permutation matrices. 3-colourable. (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 4-colourable. 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