Covers in Partitioned Intersecting Hypergraphs

TL;DR

This paper investigates covers in partitioned intersecting hypergraphs, proving conjectures related to the maximum minimal cover size for various partition configurations, including specific cases like (2,2) and (4,4).

Contribution

It confirms Tuza's conjecture for all vectors with unequal components and specific cases where components are equal, advancing understanding of hypergraph covering properties.

Findings

Proved Tuza's conjecture for b4b1b when a b1 b.

Confirmed the conjecture for b8b8 and b8b8 cases.

Established bounds on minimal cover sizes in partitioned intersecting hypergraphs.

Abstract

Given an integer $r$ and a vector $\vec{a}=(a_1, \ldots ,a_p)$ of positive numbers with $\sum_{i \le p} a_i=r$, an $r$-uniform hypergraph $H$ is said to be $\vec{a}$-partitioned if $V(H)=\bigcup_{i \le p}V_i$, where the sets $V_i$ are disjoint, and $|e \cap V_i|=a_i$ for all $e \in H,~~i \le p$. A $\vec{1}$-partitioned hypergraph is said to be $r$-partite. Let $t(\vec{a})$ be the maximum, over all intersecting $\vec{a}$-partitioned hypergraphs $H$, of the minimal size of a cover of $H$. A famous conjecture of Ryser is that $t(\vec{1})\le r-1$. Tuza conjectured that if $r>2$ then $t(\vec{a})=r$ for every two components vector $\vec{a}=(a,b)$. We prove this conjecture whenever $a\neq b$, and also for $\vec{a}=(2,2)$ and $\vec{a}=(4,4)$.