Almost partitioning every $2$-edge-coloured complete $k$-graph into $k$ monochromatic tight cycles

TL;DR

This paper proves that in any red-blue edge-coloured complete $k$-graph for $k \\geq 3$, it is possible to cover almost all vertices with $k$ disjoint monochromatic tight cycles, extending understanding of monochromatic cycle decompositions.

Contribution

It establishes a near-complete vertex coverage by $k$ disjoint monochromatic tight cycles in 2-edge-coloured complete $k$-graphs, a significant extension of previous cycle partition results.

Findings

Almost partitioning of vertices into monochromatic tight cycles.

Coverage of $n - o(n)$ vertices with $k$ cycles.

Applicable for all $k \\geq 3$ in 2-colourings.

Abstract

A $k$-uniform tight cycle is a $k$-graph with a cyclic order of its vertices such that every $k$ consecutive vertices from an edge. We show that for $k\geq 3$, every red-blue edge-coloured complete $k$-graph on $n$ vertices contains $k$ vertex-disjoint monochromatic tight cycles that together cover $n - o(n)$ vertices.