[Paper Review] Almost partitioning 2-coloured complete 3-uniform hypergraphs into two monochromatic tight or loose cycles

Sebastián Bustamante, Hiệp Hàn+1 more|arXiv (Cornell University)|Jan 26, 2017

Limits and Structures in Graph Theory14 references3 citations

TL;DR

This paper proves that for any 2-colouring of the edges of a complete 3-uniform hypergraph on sufficiently many vertices, there exist two vertex-disjoint monochromatic tight or loose cycles of different colours that cover all but at most ηn vertices, for any η > 0. The result relies on the hypergraph regularity lemma and the construction of monochromatic connected matchings in dense hypergraphs.

ABSTRACT

We show that for every η > 0 there exists an integer n_0 such that every 2-colouring of the 3-uniform complete hypergraph on n \geq n_0 vertices contains two disjoint monochromatic tight cycles of distinct colours that together cover all but at most ηn vertices. The same result holds if we replace tight cycles with loose cycles.

Motivation & Objective

• To establish an approximate version of a conjecture on covering 3-uniform hypergraphs with monochromatic tight cycles in two colours.
• To extend known results on monochromatic cycle covers from graphs to 3-uniform hypergraphs, particularly for tight and loose cycles.
• To demonstrate that in dense 2-coloured 3-uniform hypergraphs, two disjoint monochromatic connected matchings can cover almost all vertices.
• To show that the cycles can be constructed with prescribed parity, enabling the derivation of results for both tight and loose cycles.

Proposed method

• Apply the hypergraph regularity lemma to reduce the problem to a bounded-structure setting with regularity and density conditions.
• Construct monochromatic connected matchings in the reduced hypergraph by identifying connected components and using regularity properties.
• Use Lemma 3.3 to embed long tight paths in regular triads corresponding to matching edges, ensuring almost full vertex coverage.
• Build short tight cycles using non-prohibited edges between adjacent triads, ensuring disjointness via vertex set avoidance during construction.
• Replace short paths in the initial cycles with long paths of desired parity using S-avoiding constructions, preserving disjointness.
• Control error terms via bounds on the number of vertices in exceptional sets and uncovered parts, ensuring total uncovered vertices are at most ηn.

Experimental results

Research questions

• RQ1 Can every 2-coloured complete 3-uniform hypergraph be almost partitioned into two monochromatic tight cycles of different colours?
• RQ2 Does the same result hold when tight cycles are replaced by loose cycles?
• RQ3 Can the monochromatic cycles be constructed with a prescribed parity of length?
• RQ4 Is it possible to cover all but a sublinear number of vertices with two disjoint monochromatic cycles in 2-coloured 3-uniform hypergraphs?
• RQ5 What is the minimal number of vertices left uncovered in such a partition, and can this be improved beyond o(n)?

Key findings

• For every η > 0, there exists n₀ such that every 2-colouring of Kₙ⁽³⁾ with n ≥ n₀ contains two vertex-disjoint monochromatic tight cycles of different colours covering all but at most ηn vertices.
• The same conclusion holds for loose cycles, as even-length tight cycles contain loose cycles, enabling the derivation of Corollary 1.2.
• The construction allows control over the parity of the cycle lengths, which is essential for ensuring the existence of loose cycles within even-length tight cycles.
• The number of uncovered vertices is bounded by ηn, and this bound is achieved via careful application of the regularity lemma and S-avoiding path embeddings.
• The proof establishes that two disjoint monochromatic connected matchings can cover at least (1 − 290γ¹ᐟ⁶)t vertices in any 2-coloured 3-uniform hypergraph with t vertices and at least (1−γ) binom(t,3) edges.
• The error term ηn is derived from contributions of exceptional sets, uncovered parts of matchings, and the regularity framework, with total error bounded by ηn for sufficiently large n.