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Looms

Ron Aharoni, Eli Berger, Joseph Briggs, He Guo, Shira Zerbib

Abstract

A pair $(A,B)$ of hypergraphs is called orthogonal if $|a \cap b|=1$ for every pair of edges $a \in A$ and $b \in B$. An orthogonal pair of hypergraphs is called a loom if each of its two members is the set of minimum covers of the other. Looms appear naturally in the context of a conjecture of Gyárfás and Lehel on the covering number of cross-intersecting hypergraphs. We study their properties and ways of construction, and prove special cases of a conjecture that if true would imply the Gyárfás--Lehel conjecture.

Looms

Abstract

A pair of hypergraphs is called orthogonal if for every pair of edges and . An orthogonal pair of hypergraphs is called a loom if each of its two members is the set of minimum covers of the other. Looms appear naturally in the context of a conjecture of Gyárfás and Lehel on the covering number of cross-intersecting hypergraphs. We study their properties and ways of construction, and prove special cases of a conjecture that if true would imply the Gyárfás--Lehel conjecture.
Paper Structure (11 sections, 37 theorems, 34 equations)

This paper contains 11 sections, 37 theorems, 34 equations.

Table of Contents

  1. A conjecture on cross-intersecting hypergraphs, and the definition of "looms"
  2. Bounding the fractional matching number
  3. Many mutually cross-intersecting hypergraphs
  4. The fractional matching number of looms
  5. Pinnability
  6. Constructions
  7. Composition of looms
  8. Blow-ups: a generalization of composition
  9. Stars versus perfect matchings
  10. $(r,2)$-looms
  11. $(3,3)$-looms

Key Result

Theorem 1.3

If $H_1,\dots, H_m$ are $r$-uniform and $\tau(\bigcup_{i \in I}H_i)>(2r-1)(|I|-1)$ for every $I\subseteq [m]$ then $\nu_R(\mathcal{H})=m$ (namely there exists a full rainbow matching).

Theorems & Definitions (94)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Definition 1.5
  • Lemma 1.7
  • proof
  • Lemma 1.8
  • proof
  • Lemma 1.9
  • ...and 84 more