ON CONJECTURES OF BERGE AND CHVATAL

We investigate the relations among the chromatic index q(H), the maximum degree Delta(H), the total chromatic number q*(H), and the maximum size Delta(0)(H) of an intersecting subhypergraph of a hypergraph H. For some particular classes of hypergraphs, including Steiner systems, we provide sufficient conditions insuring that some (or all) of the trivial inequalities Delta(H) less than or equal to Delta(0)(H) less than or equal to q(H) less than or equal to q*(H) turn to equality. For instance, we prove that Delta(H) = Delta(0)(H) holds whenever the maximum degree of a hypergraph H is sufficiently large with respect to the rank and the 'pair-degree' of H.