A remark on the conjecture of Erdös, Faber and Lovász

Abstract.

The celebrated Erdös, Faber and Lovász Conjecture may be stated as follows: Any linear hypergraph on ν points has chromatic index at most ν. We show that the conjecture is equivalent to the following assumption: For any graph $\chi(G) \leq \nu(G)$, where ν(G) denotes the linear intersection number and χ(G) denotes the chromatic number of G. As we will see $\chi(G) + \chi(\bar{G}) \leq \nu(G)+ \nu(\overline{G})$ for any graph G  = (V, E), where $(\overline{G})$ denotes the complement of G. Hence, at least G or $\overline{G}$ fulfills the conjecture.