On the domatic and the total domatic numbers of the 2-section graph of the order-interval hypergraph of the 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 and the total domatic numbers 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)) and maximal bound of dt(G(Pl,u)) when P is the chain product Cn1×Cn2. Moreover, we give some exact values or lower bounds for d(G(P*Q)) and dt(G(Pl,u)), when * is either the direct sum or the linear sum. Finally we show that the domatic number and the total domatic number problems in this class of graphs are NP-complete.