Critical hypergraphs for the weak chromatic number

Instead of removing a vertex or an edge from a hypergraph H, one may add to some edges of H new vertices (not necessarily belonging to VH). The weak chromatic number of H tends to drop by this operation. This suggests the definition of an order relation ≥ on the set S of all Sperner hypergraphs on a universal set V of vertices. The corresponding criticality study leads to unifying and interesting results: reconstruction of critical hypergraphs and two general characterizations of k-chromatic critical hypergraphs (k ≥ 3), from which a special characterization of 3-chromatic critical hypergraphs can be derived.