Coloring linear hypergraphs: the Erd\H{o}s-Faber-Lov\'asz conjecture and the Combinatorial Nullstellensatz

TL;DR

This paper introduces an algebraic approach to the Erd ext{"o}s-Faber-Lov ext{"a}sz} conjecture on hypergraph coloring, linking it to polynomial coefficients and graph orientations, and verifies a key necessary condition.

Contribution

It formulates a stronger algebraic conjecture and connects hypergraph coloring to polynomial coefficient non-vanishing and graph orientations, advancing the theoretical understanding.

Findings

Reduced the conjecture to polynomial coefficient non-zero conditions

Established existence of certain graph orientations

Provided a necessary condition for the algebraic approach

Abstract

The long-standing Erd\H{o}s-Faber-Lov\'asz conjecture states that every $n$-uniform linear hypergaph with $n$ edges has a proper vertex-coloring using $n$ colors. In this paper we propose an algebraic framework to the problem and formulate a corresponding stronger conjecture. Using the Combinatorial Nullstellensatz, we reduce the Erd\H{o}s-Faber-Lov\'asz conjecture to the existence of non-zero coefficients in certain polynomials. These coefficients are in turn related to the number of orientations with prescribed in-degree sequences of some auxiliary graphs. We prove the existence of certain orientations, which verifies a necessary condition for our algebraic approach to work.