In this paper, we propose two contributions about the determination of chromatic number and the verification of the 2-colorability property. We introduce an unreleased relation between the problem of hypergraph coloring and the computation of minimal transversals hypergraph and, especially, a subset of them. Thereby, we propose two algorithms in order to optimize the verification of the 2-colorability property of hypergraphs and the evaluation of the chromatic number. Experiments carried out on several types of hypergraphs, showed that our algorithm obtains very interesting results.