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The Berge-Füredi conjecture on the chromatic index of hypergraphs with large hyperedges

Alain Bretto, Alain Faisant, Francois Hennecart

Abstract

This paper is concerned with two conjectures which are intimately related. The first is a generalization to hypergraphs of Vizing's Theorem on the chromatic index of a graph and the second is the well-known conjecture of Erdős, Faber and Lovász which deals with the problem of coloring a family of cliques intersecting in at most one vertex. We are led to study a special class of uniform and linear hypergraphs for which a number of properties are established.

The Berge-Füredi conjecture on the chromatic index of hypergraphs with large hyperedges

Abstract

This paper is concerned with two conjectures which are intimately related. The first is a generalization to hypergraphs of Vizing's Theorem on the chromatic index of a graph and the second is the well-known conjecture of Erdős, Faber and Lovász which deals with the problem of coloring a family of cliques intersecting in at most one vertex. We are led to study a special class of uniform and linear hypergraphs for which a number of properties are established.
Paper Structure (15 sections, 18 theorems, 36 equations, 2 figures)

This paper contains 15 sections, 18 theorems, 36 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definitions, notations and general properties
  3. Proofs and remarks
  4. When all hyperedges are large
  5. Proof of Theorem \ref{['theo15']}
  6. Proof of Theorem \ref{['theo16']}
  7. Studying the class $\mathrsfs{H}_k$
  8. General facts
  9. The cases $k=2,3$
  10. The general case
  11. Remarks and questions
  12. Further results
  13. Some consequences
  14. A generalization of Theorem \ref{['theo16']}
  15. Consequences for the Erdős-Faber-Lovász Conjecture

Key Result

Theorem 1.1

The chromatic index of any simple graph $\Gamma$ satisfies

Figures (2)

  • Figure 1: A partial hypergraph of $\mathcal{H}$ with $7$ lines immersed in the field plane $\mathcal{A}_3$.
  • Figure 4: From $\mathcal{A}_k$ to $\mathcal{H}'$. Vertices of $e_0$ and $e_1$ are labelled so that the lines $\langle x_i,y_i\rangle$ ($1\le i\le k-1$) are parallel in $\mathcal{A}_k$ and $\mathcal{H}'$.

Theorems & Definitions (35)

  • Theorem 1.1: Vizing's Theorem, 1964
  • Conjecture 1.2: Generalized Vizing’s Theorem
  • Conjecture 1.3: Erdős-Faber-Lovász
  • Conjecture 1.4: Erdős-Faber-Lovász
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 25 more