Packing tight Hamilton cycles in 3-uniform hypergraphs

TL;DR

This paper introduces new techniques to nearly perfectly pack tight Hamilton cycles in 3-uniform hypergraphs and pseudo-random digraphs, under certain conditions, advancing understanding of cycle packings in complex combinatorial structures.

Contribution

The paper develops novel methods to prove near-complete packings of tight Hamilton cycles in 3-uniform hypergraphs and pseudo-random digraphs, under natural pseudo-random conditions.

Findings

Almost all edges of certain 3-uniform hypergraphs can be covered by edge-disjoint tight Hamilton cycles.

Random 3-uniform hypergraphs can be almost completely packed with tight Hamilton cycles with high probability.

A similar packing result is established for pseudo-random digraphs with an even number of vertices.

Abstract

Let H be a 3-uniform hypergraph with N vertices. A tight Hamilton cycle C \subset H is a collection of N edges for which there is an ordering of the vertices v_1, ..., v_N such that every triple of consecutive vertices {v_i, v_{i+1}, v_{i+2}} is an edge of C (indices are considered modulo N). We develop new techniques which enable us to prove that under certain natural pseudo-random conditions, almost all edges of H can be covered by edge-disjoint tight Hamilton cycles, for N divisible by 4. Consequently, we derive the corollary that random 3-uniform hypergraphs can be almost completely packed with tight Hamilton cycles w.h.p., for N divisible by 4 and P not too small. Along the way, we develop a similar result for packing Hamilton cycles in pseudo-random digraphs with even numbers of vertices.