Monochromatic Hamiltonian Berge-cycles in colored complete uniform hypergraphs

  • Authors:

  • András Gyárfás;Jenő Lehel;Gábor N. Sárközy;R. H. Schelp

  • Affiliations:

  • Computer and Automation Research Institute, Hungarian Academy of Sciences, Budapest, PO Box 63, Budapest, Hungary H-1518;Computer and Automation Research Institute, Hungarian Academy of Sciences, Budapest, PO Box 63, Budapest, Hungary H-1518 and University of Memphis, Department of Mathematical Sciences, Memphis, TN ...;Computer and Automation Research Institute, Hungarian Academy of Sciences, Budapest, PO Box 63, Budapest, Hungary H-1518 and Computer Science Department, Worcester Polytechnic Institute, Worcester ...;University of Memphis, Department of Mathematical Sciences, Memphis, TN 38152, USA

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2008

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Abstract

We conjecture that for any fixed r and sufficiently large n, there is a monochromatic Hamiltonian Berge-cycle in every (r-1)-coloring of the edges of K"n^(^r^), the complete r-uniform hypergraph on n vertices. We prove the conjecture for r=3,n=5 and its asymptotic version for r=4. For general r we prove weaker forms of the conjecture: there is a Hamiltonian Berge-cycle in @?(r-1)/2@?-colorings of K"n^(^r^) for large n; and a Berge-cycle of order (1-o(1))n in (r-@?log"2r@?)-colorings of K"n^(^r^). The asymptotic results are obtained with the Regularity Lemma via the existence of monochromatic connected almost perfect matchings in the multicolored shadow graph induced by the coloring of K"n^(^r^).