Linear Hypergraph List Edge Coloring - Generalizations of the EFL Conjecture to List Coloring

TL;DR

This paper explores the list edge coloring of linear hypergraphs, proposing conjectures that extend classical graph coloring theorems to hypergraphs and providing partial proofs for large parameters.

Contribution

It introduces new conjectures for list edge coloring in linear hypergraphs that generalize the EFL conjecture and Vizing's theorem, with partial validation for large ranks and degrees.

Findings

Conjectures extend EFL and Vizing's theorem to hypergraphs.

Partial proofs established for large fixed rank and degree.

Proposes bounds on list edge chromatic number for hypergraphs.

Abstract

Motivated by the Erd\H{o}s-Faber-Lov\'asz (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We discuss several conjectures for list edge coloring linear hypergraphs that generalize both EFL and Vizing's theorem for graphs. For example, we conjecture that in a linear hypergraph of rank 3, the list edge chromatic number is at most 2 times the maximum degree plus 1. We show that for sufficiently large fixed rank and sufficiently large degree, the conjectures are true.