### The b-domatic number of a graph

Odile Favaron
2013 Discussiones Mathematicae Graph Theory
Besides the classical chromatic and achromatic numbers of a graph related to minimum or minimal vertex partitions into independent sets, the b-chromatic number was introduced in 1998 thanks to an alternative definition of the minimality of such partitions. When independent sets are replaced by dominating sets, the parameters corresponding to the chromatic and achromatic numbers are the domatic and adomatic numbers d(G) and ad(G). We introduce the b-domatic number bd(G) as the counterpart of the
more » ... b-chromatic number by giving an alternative definition of the maximality of a partition into dominating sets. We initiate the study of bd(G) by giving some properties and examples. Unauthenticated Download Date | 8/15/17 8:00 AM 748 O. Favaron We consider finite simple graphs G = (V, E) of order |V | = n and minimum degree δ. A set S of vertices is independent if the induced subgraph G[S] has no edge. The independence property is hereditary in the sense that every subset of an independent set is independent. A set S is dominating in G if every vertex of V \ S has at least one neighbor in S. The domination property is cohereditary in the sense that every superset of a dominating set is dominating. The minimum cardinality of a dominating set is the domination number γ(G). A dominating set is divisible if it contains two disjoint dominating sets of G, indivisible otherwise. We denote by ω(G) the maximum cardinality of a clique of G. A partition P of G is a partition of its vertex set V . Its cardinality |P| is the number of classes of P. A vertex v of G is colorful in P if v has a neighbor in each class of P different from its own class. We say that a class C of P is a-colorful if there exists an edge of G between C and every other class of P and b-colorful (called colorful in ) if it contains a colorful vertex. A partition P is proper, or chromatic, if all its classes are independent.