Transversals in Uniform Linear Hypergraphs

TL;DR

This paper investigates the bounds on the transversal number of uniform linear hypergraphs, disproves a longstanding conjecture for large uniformities, and introduces a new technique called deficiency to analyze these bounds.

Contribution

It disproves the conjecture for large k, establishes the order of the best constant in the bound, and introduces the deficiency technique for hypergraph analysis.

Findings

Disproves the conjecture for large k.

Shows the order of the best constant c_k is about ln(k)/k.

Proves the conjecture holds for k=4, setting a lower bound of 5 for k_min.

Abstract

The transversal number $\tau(H)$ of a hypergraph $H$ is the minimum number of vertices that intersect every edge of $H$. A linear hypergraph is one in which every two distinct edges intersect in at most one vertex. A $k$-uniform hypergraph has all edges of size $k$. It is known that $\tau(H) \le (n + m)/(k+1)$ holds for all $k$-uniform, linear hypergraphs $H$ when $k \in \{2,3\}$ or when $k \ge 4$ and the maximum degree of $H$ is at most two. It has been conjectured that $\tau(H) \le (n+m)/(k+1)$ holds for all $k$-uniform, linear hypergraphs $H$. We disprove the conjecture for large $k$, and show that the best possible constant $c_k$ in the bound $\tau(H) \le c_k (n+m)$ has order $\ln(k)/k$ for both linear (which we show in this paper) and non-linear hypergraphs. We show that for those $k$ where the conjecture holds, it is tight for a large number of densities if there exists an affine plane $AG(2,k)$ of order $k \ge 2$. We raise the problem to find the smallest value, $k_{\min}$, of $k$ for which the conjecture fails. We prove a general result, which when applied to a projective plane of order $331$ shows that $k_{\min} \le 166$. Even though the conjecture fails for large $k$, our main result is that it still holds for $k=4$, implying that $k_{\min} \ge 5$. The case $k=4$ is much more difficult than the cases $k \in \{2,3\}$, as the conjecture does not hold for general (non-linear) hypergraphs when $k=4$. Key to our proof is the completely new technique of the deficiency of a hypergraph introduced in this paper.