Abstract
Erdźs, Faber and Lovasz conjectured in 1972 that the vertices of a linear hypergraph with n edges, each of size n, can be strongly colored with n colors. It was shown by Romero and Sanchez-Arroyo that an equivalent conjecture is obtained when linear hypergraphs are replaced by n-clusters. In this paper we describe new families of EFL-compliant n-clusters; that is, those for which the conjecture holds. Moreover, we describe ways to extend some n-clusters to larger ones preserving EFL-compliance. Also, our approach allowed us to provide a new upper bound for the chromatic number of n-clusters.
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