Partitioning $2$-coloured complete $k$-uniform hypergraphs into monochromatic $\ell$-cycles

Abstract

We show that for all $\ell ,k,n$ with $\ell \le k/2$ and $(k-\ell )$ dividing $n$ the following hypergraph-variant of Lehel's conjecture is true. Every $2$-edge-colouring of the $k$-uniform complete hypergraph ${\mathcal{K}}_n^{(k)}$ on $n$ vertices has at most two disjoint monochromatic $\ell$-cycles in different colours that together cover all but at most $4(k-\ell )$ vertices. If $\ell \le k/3$, then at most two $\ell$-cycles cover all but at most $2(k-\ell )$ vertices. Furthermore, we can cover all vertices with at most $4$ ($3$ if $\ell \le k/3$) disjoint monochromatic $\ell$-cycles.

14 pages, 2 figures

References in corpus (3)

• Vertex covers by monochromatic pieces - A survey of results and problems
• Almost partitioning 2-coloured complete 3-uniform hypergraphs into two monochromatic tight or loose cycles
• Monochromatic loose path partitions in k-uniform hypergraphs