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Rainbow Hamilton cycles in uniform hypergraphs
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@MISC{Dudek_rainbowhamilton,
author = {Andrzej Dudek and Alan Frieze and Andrzej Ruciński},
title = {Rainbow Hamilton cycles in uniform hypergraphs},
year = {}
}
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Abstract
be the complete k-uniform hypergraph, k ≥ 3, and let ℓ be an integer such that 1 ≤ ℓ ≤ k − 1 and k − ℓ divides n. An ℓ-overlapping Hamilton cycle in K (k) n is a spanning subhypergraph C of K (k) n with n/(k − ℓ) edges and such that for some cyclic ordering of the vertices each edge of C consists of k consecutive vertices and every pair of adjacent edges in C intersects in precisely ℓ vertices. We show that, for some constant c = c(k, ℓ) and sufficiently large n, for every coloring (partition) of the edges of K (k) n which uses arbitrarily many colors but no color appears more than cn k−ℓ times, there exists a rainbow ℓ-overlapping Hamilton cycle C, that is every edge of C receives a different color. We also prove that, for some constant c ′ = c ′ (k, ℓ) and sufficiently large n, for every coloring of the edges of K (k) n in which the maximum degree of the subhypergraph induced by any single color is bounded by c ′ n k−ℓ, there exists a properly colored ℓ-overlapping Hamilton cycle C, that is every two adjacent edges receive different colors. For ℓ = 1, both results are (trivially) best possible up to the constants. It is an open question if our results are also optimal for 2 ≤ ℓ ≤ k − 1. The proofs rely on a version of the Lovász Local Lemma and incorporate some ideas from Albert, Frieze, and Reed.
