A Study Of The Upper Domatic Number Of A Graph

Given a graph G we can partition the vertices of G in to k disjoint sets. We say a set A of vertices dominates another set of vertices, B, if for every vertex in B there is some adjacent vertex in A. The upper domatic number of a graph G is written as D(G) and defined as the maximum integer k such that G can be partitioned into k sets where for every pair of sets A and B either A dominates B or B dominates A or both. In this thesis we introduce the upper domatic number of a graph and provide various results on the properties of the upper domatic number, notably that D(G) is less than or equal to the maximum degree of G, as well as relating it to other well-studied graph properties such as the achromatic, pseudoachromatic, and transitive numbers.