Linear connectivity forces large complete bipartite minors

  • Authors:

  • Thomas Böhme;Ken-ichi Kawarabayashi;John Maharry;Bojan Mohar

  • Affiliations:

  • Institut für Mathematik, Technische Universität Ilmenau, Ilmenau, Germany;Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan;Department of Mathematics, The Ohio State University, Columbus, OH, USA;Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2009

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Abstract

Let a be an integer. It is proved that for any s and k, there exists a constant N=N(s,k,a) such that every 312(a+1)-connected graph with at least N vertices either contains a subdivision of K"a","s"k or a minor isomorphic to s disjoint copies of K"a","k. In fact, we prove that connectivity 3a+2 and minimum degree at least 312(a+1)-3 are enough while the other conditions cannot be weakened. When s=1 and k=a, this implies that every 312(a+1)-connected graph with at least N(a) vertices has a K"a-minor. This is the first result where a linear lower bound on the connectivity in terms of the parameter a forces a K"a-minor. This resolves a conjecture proposed by Mader [W. Mader, Existenz n-fach zusammenhangender Teilgraphen in Graphen genugend grosser Kantendichte, Abh. Math. Sem. Univ. Hamburg 37 (1972) 86-97] and Thomason [A. Thomason, An extremal function for contractions of graphs, Math. Proc. Cambridge Philos. Soc. 95 (1984) 261-265; A. Thomason, The extremal function for complete minors, J. Combin. Theory Ser. B 81 (2001) 318-338]. Our result also generalizes a recent result of Bohme and Kostochka [T. Bohme, A. Kostochka, Disjoint K"r-minors in large graphs with given average degree, European J. Combin. 26 (2005) 289-292], resolves a conjecture of Fon-Der-Flaass [D. Fon-Der-Flaass, private communication], and has several other consequences.