Bounds on the l-total k-domatic number of a graph

Abstract

We consider finite, undirected and simple graphs G with vertex set V(G). For two integers l ≥ 0 and k ≥ 1 a set S of vertices of a graph G is an l-total k-dominating set of G, if every vertex in S has at least l neighbors in S and every vertex in V(G) - S has at least k neighbors in S. If (at least) one l-total k-dominating set exists in the graph G, then the l-total k-dominating number γl, k(G) is the minimum cardinality of such a set. An l-total k-domatic partition of G is a partition of V(G) into l-total k-dominating sets. The l-total k-domatic number d_{l, k}(G) of G is the maximum number of sets in a partition of V(G) into l-total k-dominating sets. In this work, we present bounds on the l-total k-domatic number, and we establish Nordhaus-Gaddum inequalities. Results involving the ordinary domatic number, the total domatic number, the k-domatic number and the k-tuple domatic number are improved as a consequence of this generalized approach.