On the b-domatic number of graphs

Abstract

A set of vertices S in a graph G = (V, E) is a dominating set if every vertex not in S is adjacent to at least one vertex in S. A domatic partition of graph G is a partition of its vertex-set V into dominating sets. A domatic partition P of G is called b-domatic if no larger domatic partition of G can be obtained from P by transferring some vertices of some classes of P to form a new class. The minimum cardinality of a b-domatic partition of G is called the b-domatic number and is denoted by bd(G). In this paper, we explain some properties of b-domatic partitions, and we determine the b-domatic number of some families of graphs.

Some new results on the b-domatic number of graphs

“…Note that bd(H 1 ) = 2 as proved in [5]. Likewise bd(H 2 ) = 2 by Theorem 2.3 since z is isolated in U 2 and each of a, b has a private neighbor with respect to U 1 .…”

“…In [5], the authors gave some sufficient conditions for graphs to attain equality in the bound of Proposition 2.2. Recall that a set S ⊆ V is independent if no two vertices in S are adjacent.…”

“…If G has a vertex whose neighbors form an independent set, then bd(G) = 2. Proposition 2.4 ( [5]). If G is a prism graph, then bd(G) = 2.…”

“…It has been shown in [5] that if G has a universal vertex v, then bd(G v) = bd(G) − 1. This result can be generalized as follows.…”

“…In the sequel, we shall show that all graphs of H, except those depicted in Figure 1, have a b-domatic number equal to 3. Recall that it was shown in [5] that bd(H 1 ) = bd(H 4 ) = 2. Let U 1 = {a, b} and U 2 = {x, y, z, t}.…”