Date of Award
4-1984
Degree Name
Doctor of Philosophy
Department
Mathematics
First Advisor
Dr. Linda Lesniak
Second Advisor
Dr. Gary Chartrand
Third Advisor
Dr. John Petro
Abstract
Clearly, there are many ways that one can partition the vertex sets of graphs. In the first chapter of this work I examine the problem of determining, for a given graph, the minimum order of a vertex partition having specified properties. In the remaining chapters I concentrate on partitions of two types--those in which each subset induces an empty graph and those in which each subset induces an empty or a complete graph.
The chromatic number of a graph G is the minimum number of subsets into which V(G) can be partitioned so that each subset induces an empty graph. The cochromatic number of G is the minimum number of subsets into which V(G) can be partitioned so that each subset induces a complete or an empty graph. In the second chapter I discuss the relationship between chromatic and cochromatic numbers of graphs. I also extend known results in the field of cochromatic numbers.
In the third chapter I explore concepts in cochromatic theory which are analogous to well known topics in chromatic theory.
The acochromatic number of a graph G is the maximum order of all vertex partitions of G where each subset induces a complete or an empty graph but the union of any two does neither. I show in Chapter IV that the acochromatic number of a bipartite graph is bounded below by its edge independent number and above by this number plus one.
In the last chapter I discuss switching sets and sequences. I apply knowledge of chromatic and cochromatic theory to this concept.
Access Setting
Dissertation-Open Access
Recommended Citation
Gimbel, John Gordon, "The Chromatic and Cochromatic Number of a Graph" (1984). Dissertations. 2371.
https://scholarworks.wmich.edu/dissertations/2371