Monochromatic Hamiltonian Berge-cycles in colored hypergraphs

TL;DR

This paper proves the existence of monochromatic Hamiltonian Berge-cycles in large colored hypergraphs, extending known results and confirming a conjecture for specific cases with fewer colors.

Contribution

It demonstrates the conjecture holds with r-2 colors and provides a proof for the case r=4, advancing understanding of monochromatic cycles in hypergraphs.

Findings

Existence of monochromatic Hamiltonian t-tight Berge-cycles in large hypergraphs with fewer colors

Proof of the conjecture for r=4 case

Improved bounds over previous results

Abstract

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 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 b r-2 / t-1 -edge coloring of Kr n 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 [2, 3, 4].