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 d t ( G ( P ) ) 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 ) ) when P is the chain product C n 1 × C n 2 . According to the values of l , u , n 1 , n 2 , the maximal domatic partition is exhibited. Moreover, we give some exact values or lower bounds for d ( G ( P ∗ Q ) ) and d t ( G ( P l , 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.