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Graph Theory: Extremal Problem

  1. Apr 7, 2014 #1
    1. The problem statement, all variables and given/known data

    Homework, from Modern Graph Theory by Bela Bollobas, section on extremals:

    1. Suppose that G is a graph with n > r + 1 vertices and tr(n) + 1 edges.
    (a) Prove that for every p with r + 1 < p <= n there is a subgraph H of G
    with |H| = p and e(H) >= tr(p) + 1. [Hint: Try to copy the proof of Turan’s
    Theorem. You may wish to write n = qr + x where 0 <= x < r, and consider
    the cases x = 0, x = 1 and x > 2.]
    (b) Prove that G contains two copies of Kr+1 with exactly r common vertices


    2. Relevant equations

    tr(n) − δ(Tr(n)) = tr(n − 1).
    where tr(n) is the Turan number, Tr(n) the Turan graph, and δ(G) is the minimum degree of G

    3. The attempt at a solution

    Use a descending induction on p.

    Assume there exists a subgraph of order p and size greater or equal to tr(p) + 1.
    Identify a vertex v with deg(v) <= δ(Tr(p)) = p - roof(p/r)
    Delete v and its incident vertices
    Since tr(p) − δ(Tr(p)) = tr(p − 1) the result is a subgraph of order p-1 and size greater or equal to tr(p-1) + 1 as required.
    The result follows by induction.

    The crucial step is proving the existence of a suitable v, which I've no clue how to do.
     
    Last edited: Apr 7, 2014
  2. jcsd
  3. Apr 8, 2014 #2
    Still having trouble showing that there is a vertex with degree less or equal to δ(Tr(p))
     
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