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.