Alon-Tarsi for hypergraphs

2024-12-30

AI Summary

This work investigates the quantitative relationship between the hypergraph Alon–Tarsi number ( mathrm{AT}(p_H) ) and the edge density ( mathrm{ed}(H) ), aiming to establish a hypergraph generalization of the 1-2-3 Conjecture. Method: We construct, for each hypergraph ( H ), a product polynomial ( p_H )—the first systematic extension of the Alon–Tarsi method to hypergraphs—and apply algebraic combinatorics, signed permutation analysis, and the polynomial method (via zero-set arguments). Contribution/Results: When ( p_H ) has unit coefficients, we prove ( mathrm{AT}(p_H) = lceil mathrm{ed}(H) ceil + 1 ); for arbitrary coefficients, coefficient rearrangement yields the tight upper bound ( mathrm{AT}(p'_H) leq 2lceil mathrm{ed}(H) ceil + 1 ). We propose the central conjecture that ( mathrm{AT}(p_H) leq 2lceil mathrm{ed}(H) ceil + 1 ) holds *without* rearrangement—its validity would directly imply both the 1-2-3 Conjecture and its weighted hypergraph extension. Our framework establishes a novel algebraic paradigm for list coloring and weighted edge coloring of hypergraphs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Given a hypergraph $H=(V,E)$, define for every edge $ein E$ 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 $AT(p_H)=lceil ed(H) ceil+1$ 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 $p_H^prime$, $AT(p_H^prime)leq 2lceil ed(H) ceil+1$ 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.

Problem

• Hypergraph p_H Alon-Tarsi numbers
• Line density
• 1-2-3 Conjecture

Innovation

• Alon-Tarsi number
• hypergraph edge density
• 1-2-3 conjecture advancement