PS02-100205

# PS02-100205 - (b(5 points Using the above recurrence ﬁnd...

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MATH 682 Problem Set #2 This problem set is due at the beginning of class on February 4 . Below, “graph” means “simple ﬁnite graph”. 1. (10 points) Let ˜ K n,n be the bipartite graph with vertex set { a 1 ,a 2 ,...,a n ,b 1 ,b 2 ,...,b n } and containing the edge { a i ,b j } if and only if i 6 = j . (a) (5 points) Show that ˜ K n,n satisﬁes the Hall marriage criterion and thus has a perfect matching. (b) (5 points) How many diﬀerent perfect matchings are there on ˜ K n,n (hint: ﬁnd the number of perfect matchings on K n,n , and exclude those which match some a i with b i ). 2. (15 points) Let f ( G ) be the number of (not necessarily perfect) matchings on a graph G . (a) (5 points) Show that for any edge e = { u,v } in G , f ( G ) = f ( G - e )+ f ( G - u - v ), where G - e represents G with the edge e removed, and G - v represents G with the vertex v and all incident edges removed, and that f ( G ) = 1 if k G k = 0.
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Unformatted text preview: (b) (5 points) Using the above recurrence, ﬁnd the number of matchings on the following graph: (c) (5 points) How could the above recurrence be modiﬁed to give the number of perfect matchings? 3. (10 points) Prove Hall’s Theorem by restricting Tutte’s Theorem to the bipartite case and exhibiting that the Hall criterion follows from the Tutte criterion if G is bipartite. 4. (5 points) Show without using Menger’s Theorem that if G is 2-connected and u and v are distinct vertices of G , there is a cycle in G containing both u and v . A man is about thirty-eight before he stockpiles enough socks to be able to get one matching pair. —Merrily Harpur Page 1 of 1 Due February 4, 2010...
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