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Paper   IPM / M / 14712
School of Mathematics
  Title:   Monochromatic Hamiltonian Berge-cycles in colored hypergraphs
  Author(s):  G. R. Omidi (Joint with L. Maherani)
  Status:   Published
  Journal: Discrete Math.
  Vol.:  34
  Year:  2017
  Pages:   2043-2052
  Supported by:  IPM
It has been conjectured 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 Krn, the complete r-uniform hypergraph on n vertices. In this paper, we show that the statement of this conjecture is true with r − 2 colors (instead of r − 1 colors) by showing that there is a monochromatic Hamiltonian t-tight Berge-cycle in every ⌊ r−2 t−1 ⌋-edge-coloring of Krn for any fixed r > t ≥ 2 and sufficiently large n. Also, we give a proof for this conjecture when r = 4 (the first open case). These results improve the previously known results in Dorbec et al. (2008) and Gyárfás et al. (2008, 2010).

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