#### Title

On Hamilton cycle decompositions of complete multipartite graphs which are both cyclic and symmetric

#### Date of Award

2021

#### Degree Name

Mathematics

#### College

College of Science

#### Type of Degree

M.A.

#### Document Type

Thesis

#### First Advisor

Dr. Michael Schroeder, Committee Chairperson

#### Second Advisor

Dr. Elizabeth Niese

#### Third Advisor

Dr. JiYoon Jung

#### Abstract

Let G be a graph with *v* vertices. A Hamilton cycle of a graph is a collection of edges which create a cycle using every vertex. A Hamilton cycle decomposition is cyclic if the set of cycle is invariant under a full length permutation of the vertex set. We say a decomposition is symmetric if all the cycles are invariant under an appropriate power of the full length permutation. Such decompositions are known to exist for complete graphs and families of other graphs. In this work, we show the existence of cyclic *n*-symmetric Hamilton cycle decompositions of a family of graphs, the complete multipartite graph K_{m×n} where the number of parts, *m*, is odd and the part size, *n*, is also odd. We classify the existence where m is prime and prove the existence in additional cases where *m* is a composite odd integer.

#### Subject(s)

Graph theory.

Decomposition (Mathematics)

#### Recommended Citation

Akinola, Fatima A., "On Hamilton cycle decompositions of complete multipartite graphs which are both cyclic and symmetric" (2021). *Theses, Dissertations and Capstones*. 1356.

https://mds.marshall.edu/etd/1356