Bounds on Erd{\H{o}}s - Faber - Lov\'{a}sz Conjecture - the Uniform and Regular Cases

TL;DR

This paper establishes upper bounds on the chromatic number for certain regular linear hypergraphs related to the Erdős-Faber-Lovász conjecture, providing new insights into hypergraph coloring.

Contribution

It introduces specific upper bounds for the chromatic number of r-regular linear hypergraphs, advancing understanding of the EFL conjecture in uniform and regular cases.

Findings

For r ≥ 4, χ(H) ≤ 1.181n

For r = 3, χ(H) ≤ 1.281n

Provides bounds relevant to the EFL conjecture

Abstract

We consider the Erd{\H{o}}s - Faber - Lov\'{a}sz (EFL) conjecture for hypergraphs. This paper gives an upper bound for the chromatic number of $r$ regular linear hypergraphs $\textbf{H}$ of size $n$. If $r \ge 4$, $\chi (\textbf{H}) \le 1.181n$ and if $r=3$, $\chi(\textbf{H}) \le 1.281n$