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 P l , u of P induced by consecutive levels ∪ i = l u N i of P, we give exact values of d ( G ( P l , u ) ) and maximal bound of d t ( G ( P l , u ) ) when P is the chain product C n 1 × C n 2 . Moreover, we give some exact values or lower bounds for d ( G ( P * Q ) ) and d t ( G ( P l , 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.