Looms

TL;DR

This paper investigates the properties and construction methods of looms, special pairs of hypergraphs related to a conjecture on hypergraph covering numbers, and proves special cases that support the conjecture.

Contribution

It introduces the concept of looms, explores their properties and construction, and proves special cases of a conjecture linking looms to the Gyárfás-Lehel conjecture.

Findings

Looms are characterized as pairs of hypergraphs with specific minimal cover properties.

Construction methods for looms are developed and analyzed.

Special cases of the conjecture relating looms to hypergraph covering numbers are proved.

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 Gy\'arf\'as 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 Gy\'arf\'as--Lehel conjecture.