Alon-Tarsi for hypergraphs

TL;DR

This paper explores the relationship between the Alon-Tarsi number of a polynomial associated with a hypergraph and its edge density, proposing bounds and conjecturing a potential generalization of the 1-2-3 Conjecture.

Contribution

It establishes bounds on the Alon-Tarsi number for hypergraph polynomials and introduces a conjecture that could extend the 1-2-3 Conjecture.

Findings

AT(p_H)=⎡ed(H)⎤+1 when coefficients are all ones

Permuting coefficients can bound AT(p_H') by 2⎡ed(H)⎤+1

Conjecture that coefficient permutation may be unnecessary

Abstract

Given a hypergraph $H=(V,E)$, define for every edge $e\in E$ a linear expression with arguments corresponding with the vertices. Next, let the polynomial $p_H$ be the product of such linear expressions for all edges. Our main goal was to find a relationship between the Alon-Tarsi number of $p_H$ and the edge density of $H$. We prove that $AT(p_H)=\lceil ed(H)\rceil+1$ if all the coefficients in $p_H$ are equal to $1$. Our main result is that, no matter what those coefficients are, they can be permuted within the edges so that for the resulting polynomial $p_H^\prime$, $AT(p_H^\prime)\leq 2\lceil ed(H)\rceil+1$ holds. We conjecture that, in fact, permuting the coefficients is not necessary. If this were true, then in particular a significant generalization of the famous 1-2-3 Conjecture would follow.