Complete Minors In K s,s -Free Graphs

We prove that for a fixed integer s≥2 every K s,s -free graph of average degree at least r contains a K p minor where \( p = r^{{1 + \frac{1} {{2{\left( {a - 1} \right)}}} + o{\left( 1 \right)}}} \). A well-known conjecture on the existence of dense K s,s -free graphs would imply that the value of the exponent is best possible. Our result implies Hadwiger’s conjecture for K s,s -free graphs whose chromatic number is sufficiently large compared with s.

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Correspondence to Daniela Kühn.

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Kühn, D., Osthus, D. Complete Minors In K s,s -Free Graphs. Combinatorica 25, 49–64 (2004).

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Mathematics Subject Classification (2000):

  • 05C83
  • 05C35
  • 05D40