## Date of Award

2016

## Document Type

Open Access Dissertation

## Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

## Administrative Home Department

Department of Mathematical Sciences

## Advisor 1

Donald L. Kreher

## Advisor 2

Melissa S. Keranen

## Committee Member 1

Stefaan De Winter

## Committee Member 2

Dalibor Froncek

## Abstract

Gerhard Ringel was an Austrian Mathematician, and is regarded as one of the most influential graph theorists of the twentieth century. This work deals with two problems that arose from Ringel's research: the Hamilton-Waterloo Problem, and the problem of R-Sequences.

The Hamilton-Waterloo Problem (HWP) in the case of C_{m}-factors and C_{n}-factors asks whether K_{v}, where v is odd (or K_{v}-F, where F is a 1-factor and *v* is even), can be decomposed into *r* copies of a 2-factor made entirely of *m*-cycles and *s* copies of a 2-factor made entirely of *n*-cycles. Chapter 1 gives some general constructions for such decompositions and apply them to the case where *m*=3 and *n*=3x. This problem is settle for odd *v*, except for a finite number of *x* values. When *v* is even, significant progress is made on the problem, although open cases are left. In particular, the difficult case of *v* even and *s*=1 is left open for many situations.

Chapter 2 generalizes the Hamilton-Waterloo Problem to complete equipartite graphs K_{(n:m)} and shows that K_{(xyzw:m)} can be decomposed into *s* copies of a 2-factor consisting of cycles of length *xzm* and *r* copies of a 2-factor consisting of cycles of length *yzm*, whenever *m* is odd, *s*,*r*≠1, gcd(*x*,*z*)=gcd(*y*,*z*)=1 and *xyz*≠0 (mod 4). Some more general constructions are given for the case when the cycles in a given two factor may have different lengths. These constructions are used to find solutions to the Hamilton-Waterloo problem for complete graphs.

Chapter 3 completes the proof of the Friedlander, Gordon and Miller Conjecture that every finite abelian group whose Sylow 2-subgroup either is trivial or both non-trivial and non-cyclic is R-sequenceable. This settles a question of Ringel for abelian groups.

## Recommended Citation

Pastine, Adrian, "Two Problems of Gerhard Ringel", Open Access Dissertation, Michigan Technological University, 2016.

*Copyright Agreement for Chapter 1*