Adomatic and idomatic numbers of graphs
Bohdan Zelinka · Czech digital mathematics library · 1983
ADOMATIC AND IDOMATIC NUMBERS OF GRAPHSBOHDAN ZELINKA E. J. Cockayne and S. T. Hedetniemi [1] have defined the domatic number of a graph and also some related concepts, among others the adomatic number of a graph and the idomatic one.Here we shall present some results concerning adomatic and idomatic numbers.We consider finite undirected graphs without loops and multiple edges.First we shall give definitions.Aall of whose classes are dominating sets in G, is called a domatic partition of G.The maximum number of classes of a domatic partition of G is called the domatic number of G and denoted by d(G).The minimum number of classes of a partition of V(G), all of whose classes are indivisible dominating sets in G, is called the adomatic number of G and denoted by ad(G).If there exists at least one domatic partition of G, all of whose classes are independent sets, then the maximum number of classes of such a partition is called the idomatic number of G and denoted by id(G).If no such partition exists, we put id(G) = 0.A graph G for which id(G)^0 is called idomatic.First we prove some assertions concerning the adomatic number.