Linear Hypergraph Edge Coloring - Generalizations of the EFL Conjecture

Motivated by the Erdos-Faber-Lovász (EFL) conjecture for hypergraphs, we consider the edge coloring of linear hypergraphs. We discuss several conjectures for 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 edge chromatic number is at most 2 times the maximum degree unless the hypergraph is the Fano plane where the number is 7. We show that for fixed rank sufficiently large and sufficiently large degree, the conjectures are true.