Date of Award:

5-2026

Document Type:

Thesis

Degree Name:

Master of Science (MS)

Department:

Mathematics and Statistics

Committee Chair(s)

Aysel Erey

Committee

Aysel Erey

Committee

David Brown

Committee

Brent Thomas

Committee

Zhaohu Nie

Abstract

The content of this thesis may be compared to a stack of plates with a common design that are broken, one at a time. If the plates are broken into large enough pieces you would be able to choose one fragment from each broken plate to discover the entire design that the plates share. For example, if you were still missing the center of the design after taking a piece from some plates, you could look for the piece of the next broken plate with the region in question.

We study the question of how many broken plates might guarantee that you can recover the original plate design–though instead of plates, we study combinatorial graphs. A graph G is a collection of things (vertices) and a collection of pairs of vertices that are connected (edges). A set of vertices D in a graph is said to be a dominating set if each vertex not in D is connected to at least one vertex in D.

Rather than how many broken plates might be needed to reconstruct a design, we consider how many dominating sets of G are sufficient to guarantee that a (potentially new) dominating set may be constructed by choosing a single vertex from each dominating set provided. We define the number of dominating sets which may be necessary for this construction to be the rainbow domination number of G, R(G). In this context, the term rainbow implies that one may think of each dominating set (or plate) given having a different color, and the newly constructed dominating set using at most one vertex (piece) of each color.

In this thesis, we define R(G) and study it for several families of graphs. We show how it is affected by several graph operators as they act on G and provide general bounds which relate R(G) to various other graph invariants. We provide examples to demonstrate the tightness of these bounds and pose several open problems and conjectures that we believe are accessible to new and upcoming researchers.

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

Download

Included in

Mathematics Commons

Share

COinS

Copyright for this work is retained by the student. If you have any questions regarding the inclusion of this work in the Digital Commons, please email us at .