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Alon-Tarsi for hypergraphs

Marcin Anholcer, Bartłomiej Bosek, Grzegorz Gutowski, Michał Lasoń, Jakub Przybyło, Oriol Serra, Michał Tuczyński, Lluís Vena, Mariusz Zając

TL;DR

The paper develops an Alon–Tarsi framework for hypergraphs by studying hypergraph polynomials $p_H$ and their Alon–Tarsi numbers $AT(p_H)$ in relation to edge density $ed(H)$. It proves that when edge coefficients may be permuted to form $p_H'$, the Alon–Tarsi bound satisfies $AT(p_H')\le 2\lceil ed(H)\rceil+1$, and conjectures that this bound may hold without any coefficient permutation. This result yields immediate corollaries bounding the list chromatic number and online choosability: $\chi_L(H)\le \chi_P(H)\le 2\lceil ed(H)\rceil+1$, and it provides a pathway toward a broad generalization of the 1-2-3 Conjecture. The paper also discusses tightness, gives concrete coefficient-permutation examples where $AT$ changes, and outlines two proofs of the main theorem, along with a program of open problems and conjectures that link hypergraph algebro-combinatorics to classical graph coloring questions.

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.

Alon-Tarsi for hypergraphs

TL;DR

The paper develops an Alon–Tarsi framework for hypergraphs by studying hypergraph polynomials and their Alon–Tarsi numbers in relation to edge density . It proves that when edge coefficients may be permuted to form , the Alon–Tarsi bound satisfies , and conjectures that this bound may hold without any coefficient permutation. This result yields immediate corollaries bounding the list chromatic number and online choosability: , and it provides a pathway toward a broad generalization of the 1-2-3 Conjecture. The paper also discusses tightness, gives concrete coefficient-permutation examples where changes, and outlines two proofs of the main theorem, along with a program of open problems and conjectures that link hypergraph algebro-combinatorics to classical graph coloring questions.

Abstract

Given a hypergraph , define for every edge a linear expression with arguments corresponding with the vertices. Next, let the polynomial be the product of such linear expressions for all edges. Our main goal was to find a relationship between the Alon-Tarsi number of and the edge density of . We prove that if all the coefficients in are equal to . Our main result is that, no matter what those coefficients are, they can be permuted within the edges so that for the resulting polynomial , 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.
Paper Structure (5 sections, 15 theorems, 77 equations)

This paper contains 5 sections, 15 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main result
  4. Applications
  5. Remarks and open problems

Key Result

Theorem 1

For every hypergraph $H=(V,E)$ and fully unbalanced hypergraph polynomial $p$ of $H$ there exist permutations of edges $\sigma_{e_1}\!\in S_{e_1}$$\sigma_{e_2}\!\in S_{e_2},$$\ldots,$$\sigma_{e_m}\!\in S_{e_m}$ such that

Theorems & Definitions (45)

  • Theorem 1
  • Example 1
  • Example 2
  • Conjecture 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Example 3
  • Lemma 3
  • ...and 35 more