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Midterm Exam for First-Year Interest Group Seminar | N 1, Exams of Health sciences

Material Type: Exam; Class: FIRST-YEAR INTEREST GROUP SMNR; Subject: Nursing; University: University of Texas - Austin; Term: Fall 2005;

Typology: Exams

Pre 2010

Uploaded on 08/31/2009

koofers-user-yf9-1
koofers-user-yf9-1 🇺🇸

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CS388C: Combinatorics and Graph Theory October 25, 2005 David Zuckerman

Midterm

Instructions. There are five problems of equal weight. You may assume any theorems stated in class or in the book, unless the question is to prove such a theorem. Explain all answers. You are allowed one sheet of paper, with writing on both sides. You have 75 minutes.

Name:

  1. Show that any family of more than 2n−^1 subsets of [n] contains distinct sets A, B with A ⊂ B.
  1. Show that any graph on 2n vertices with all degrees at least n has a perfect matching. (Hint: consider short augmenting paths.)
  1. Show that there exists a constant c with the following property. Let M be an n×n matrix with ±1 entries. Then there exists a vector v with ±1 entries such that at least 99% of the entries in M v are at most c

n in absolute value.