Graph theory, Spring-summer 2008, Exam format

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					         Graph theory, Spring-summer 2008, Exam format


   The exam will consist of two parts. In part one you will be asked to do one out of two
questions taken from section 1 below (50 points). In part two you will be asked to do two
out of three questions taken from section 2 below (25 points each). The exam will last three
hours. To avoid guesswork on your part as to which questions will appear in mo’ed Aleph
and Bet, the way in which I will choose the questions will be totally random. Reminder:
Mo’ed Aleph is August 7th and Mo’ed Bet August 28th. The exams will be at 10:00 a.m.,
we will notify you of the room. Behatzlacha raba.


1
    1. State and prove Tutte’s theorem

    2. State and prove the Matrix Tree Theorem, and deduce Cayley’s theorem from it.

    3. State and prove the mincut-maxflo theorem.

    4.    • State Szemeredi’s regularity lemma
          • Prove: For every > 0 there exists a δ > 0 such that the following holds: if
            G = (V, E) is a graph on n vertices with less than δn3 triangles, then one can
            remove n2 edges from G and destroy all triangles.

    5. Prove that there exist graphs with arbitrarily large girth and chromatic number.

    6. Define Shannon capacity and find, with proof, the Shannon capacity of C5 .

    7. Prove that every planar graph is 5-choosable. (A good reference for the proof: ”Proofs
       from the book”.)




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    8. Prove that there exist a positive constant K such that the following holds. Let A, B, C
       be sets of s real numbers each. Then
                      |A · B + C| = |{ab + c : a ∈ A, b ∈ B, c ∈ C}| ≥ Ks3/2 .
       (A good reference for the proof: ”The Probabilistic Method/ Alon and Spencer.”)


2
    1. Let T be a graph on n vertices. Use linear algebra to prove that any two of the following
       imply the third:
        (i) |E(T )| = n − 1
       (ii) T is connected.
       (iii) T has no cycles.
    2. Prove that if there exist two vertices in a graph, with two different paths between them
       then the graph contains a cycle.
    3. Prove: Every finite 2k-regular graph contains a 2-factor.
    4. Prove: Every bridgeless 3-regular graph contains a 1-factor.
    5. Let µ1 and µ2 be measures on a finite partially ordered set. Define stochastic domina-
       tion and monotone coupling, and show that µ1 stochastically dominates µ2 iff there is
       a monotone coupling between them.
    6. Prove that Mincut-Maxflow implies Dilworth.
    7. Prove that Dilworth implies Hall.
    8. Prove that any graph on n vertices has a cut (X, Y ) where |X| − |Y | is 0 or 1 and the
       cut contains strictly more than half the edges in the graph.
    9. State and prove Turan’s theorem
10. Show there exists constants b, B such that bn3/2 ≤ Ex(n, C4 ) ≤ Bn3/2 .
11. Prove: for all d, r and s there exist t and such that the following holds: If G = (V, E)
    is a graph and V = V1 ∪ V2 ∪ . . . ∪ Vr where all Vi have size at least t, and every pair
    of sets (Vi , Vj ) spans an regular graph of density at least d then G contains a copy of
    the complete r-partite graph Ks,s,...,s .

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12.    • State Szemeredi’s regularity lemma
      • State a theorem that follows from Sz.R.L. and use it to prove Roth’s theorem.
                                                                 √ k
13. Prove that for k > 3 the Ramsey number R(k, k) is such that 2 ≤ R(k, k) ≤ 4k .

14. Prove that R(3, k) = O(k 2 / log k)

15. Let G be a connected d-regular graph. Show that G’s maximal eigenvalue is d, and
    that this eigenvalue has multiplicity 1. Show that −d is an eigenvalue iff G is bi-partite.
    What happens if G is not connected?

16. Define the graph of the n-dimensional Hamming cube, find its eigenvalues and eigen-
    vectors, and use them to find the maximal independent sets in this graph. (It is easy
    to find these sets directly...)

17. Show that K5 and K3,3 are not planar.

18. Prove that there exist graphs with chromatic number 2 and arbitrarily large choice
    number.




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