Towards Lehel's conjecture for 4-uniform tight cycles

Allan Lo; Vincent Pfenninger

doi:10.37236/10604

Towards Lehel's conjecture for 4-uniform tight cycles

Allan Lo
, Vincent Pfenninger

Mathematics

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Abstract

A k-uniform tight cycle is a k-uniform hypergraph with a cyclic ordering of its vertices such that its edges are all the sets of size k formed by k consecutive vertices in the ordering. We prove that every red-blue edge-coloured K_n⁽⁴⁾ contains a red and a blue tight cycle that are vertex-disjoint and together cover n−o(n) vertices. Moreover, we prove that every red-blue edge-coloured K_n⁽⁵⁾ contains four monochromatic tight cycles that are vertex-disjoint and together cover n−o(n) vertices.

Original language	English
Article number	P1.13
Number of pages	36
Journal	Electronic Journal of Combinatorics
Volume	30
Issue number	1
DOIs	https://doi.org/10.37236/10604
Publication status	Published - 13 Jan 2023

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title = "Towards Lehel's conjecture for 4-uniform tight cycles",

abstract = "A k-uniform tight cycle is a k-uniform hypergraph with a cyclic ordering of its vertices such that its edges are all the sets of size k formed by k consecutive vertices in the ordering. We prove that every red-blue edge-coloured Kn(4) contains a red and a blue tight cycle that are vertex-disjoint and together cover n−o(n) vertices. Moreover, we prove that every red-blue edge-coloured Kn(5) contains four monochromatic tight cycles that are vertex-disjoint and together cover n−o(n) vertices.",

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AB - A k-uniform tight cycle is a k-uniform hypergraph with a cyclic ordering of its vertices such that its edges are all the sets of size k formed by k consecutive vertices in the ordering. We prove that every red-blue edge-coloured Kn(4) contains a red and a blue tight cycle that are vertex-disjoint and together cover n−o(n) vertices. Moreover, we prove that every red-blue edge-coloured Kn(5) contains four monochromatic tight cycles that are vertex-disjoint and together cover n−o(n) vertices.

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Towards Lehel's conjecture for 4-uniform tight cycles

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