Strong domatic number of a graph

Abstract

A set D of vertices of a simple graph G = (V, E) is a strong dominating set, if for every vertex x ∈ D = V \ D there is a vertex y ∈ D with xy ∈ E(G) and deg(x) ≤ deg(y). The strong domination number γ st (G) is defined as the minimum cardinality of a strong dominating set. The strong domatic number of G is the maximum number of strong dominating sets into which the vertex set of G can be partitioned. We initiate the study of the strong domatic number, and we present different sharp bounds on d st (G). In addition, we determine this parameter for some classes of graphs, such as cubic graphs of order at most 10.