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About Berge-Füredi's conjecture on the chromatic index of hypergraphs

Alain Bretto, Alain Faisant, François Hennecart

Abstract

We show that the chromatic index of a hypergraph $\mathcal{H}$ satisfies Berge-Füredi conjectured bound $\mathrm{q}(\mathcal{H})\le Δ([\mathcal{H}]_2)+1$ under certain hypotheses on the antirank $\mathrm{ar}(\mathcal{H})$ or on the maximum degree $Δ(\mathcal{H})$. This provides sharp information in connection with Erdős-Faber-Lovász Conjecture which deals with the coloring of a family of cliques that intersect pairwise in at most one vertex.

About Berge-Füredi's conjecture on the chromatic index of hypergraphs

Abstract

We show that the chromatic index of a hypergraph satisfies Berge-Füredi conjectured bound under certain hypotheses on the antirank or on the maximum degree . This provides sharp information in connection with Erdős-Faber-Lovász Conjecture which deals with the coloring of a family of cliques that intersect pairwise in at most one vertex.
Paper Structure (6 sections, 4 theorems, 16 equations)

This paper contains 6 sections, 4 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Notations and statements
  3. Proof of Theorem \ref{['thm1']}
  4. Proof of Theorem \ref{['thm2']}
  5. Proof of Theorem \ref{['thm3']}
  6. Remarks and questions

Key Result

Theorem 1

Let $\mathcal{H}=(V,E)$ be a loopless hypergraph such that $\mathrm{ar}(\mathcal{H}) \ge \sqrt{\Delta([\mathcal{H}]_2)+1}$. Then $\mathrm{q}(\mathcal{H})\le \Delta([\mathcal{H}]_2)+1$.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4: cf. Lemma 3.2 of BFH2024