Towards Lehel's conjecture for 4-uniform tight cycles

TL;DR

This paper proves that in large complete 4-uniform and 5-uniform hypergraphs with red-blue edge colourings, there exist large vertex-disjoint monochromatic tight cycles covering almost all vertices, advancing understanding of hypergraph cycle structures.

Contribution

It establishes near-complete vertex coverage by monochromatic tight cycles in 4-uniform and 5-uniform hypergraphs, extending Lehel's conjecture to these hypergraph settings.

Findings

Existence of large vertex-disjoint monochromatic tight cycles in 4-uniform hypergraphs.

Existence of four such cycles covering almost all vertices in 5-uniform hypergraphs.

Progress towards hypergraph analogues of classical cycle covering conjectures.

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.