On the domatic and the total domatic numbers of the 2-section graph of the order-interval hypergraph of a finite poset

Let P be a finite poset and H(P) be the hypergraph whose vertices are the points of P and whose edges are the maximal intervals in P. We study the domatic number d(G(P)) and the total domatic number dt(G(P)) of the 2-section graph G(P) of H(P). For the subset Pl,u of P induced by consecutive levels ∪i=luNi of P, we give exact values of d(G(Pl,u)) when P is the chain product Cn1×Cn2. According to the values of l,u,n1,n2, the maximal domatic partition is exhibited. Moreover, we give some exact values or lower bounds for d(G(P∗Q)) and dt(G(Pl,u)), when ∗ is the direct sum, the linear sum or the Cartesian product. Finally we show that the domatic number and the total domatic number problems in this class of graphs are NP-complete.