# Partitioning edge-coloured hypergraphs into few monochromatic tight cycles

## TL;DR

This paper proves that in any edge-coloured complete hypergraph or graph, vertices can be partitioned into a bounded number of monochromatic tight cycles or powers of cycles, confirming conjectures and extending previous results.

## Contribution

It establishes a unified theorem that generalizes partitioning into monochromatic cycles for hypergraphs and graphs, extending to hypergraphs with bounded independence number.

## Key findings

- Vertices of any r-edge-coloured complete hypergraph can be partitioned into a bounded number of monochromatic tight cycles.
- Vertices of any r-edge-coloured complete graph can be partitioned into a bounded number of p-th powers of cycles.
- The results extend to all host hypergraphs with bounded independence number.

## Abstract

Confirming a conjecture of Gy\'arf\'as, we prove that, for all natural numbers $k$ and $r$, the vertices of every $r$-edge-coloured complete $k$-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for for all natural numbers $p$ and $r$, the vertices of every $r$-edge-coloured complete graph can be partitioned into a bounded number of $p$-th powers of cycles, settling a problem of Elekes, Soukup, Soukup and Szentmikl\'ossy. In fact we prove a common generalisation of both theorems which further extends these results to all host hypergraphs of bounded independence number.