#### Date of Award

2008

#### Document Type

Thesis

#### Degree Name

Master of Science (MS)

#### Department

Mathematics

#### Abstract

Tom Johnson is an American musician and composer living in Paris who regularly composes pieces based on combinatorial designs. He recently posed the following question: is there a way to construct five Room squares of side seven such that each of the thirty-five possible fill patterns is used exactly once? In this thesis we study his question and several related questions concerning the possible fill patterns in sets of Room squares of side 7 and 11. The second problem addressed in this thesis is a conjecture by Marco Buratti, a combinatorialist from the Universitia di Perusia, Italy. He conjectured that given p a prime and a multiset S containing p - 1 non-zero elements from Zp, there exists a Hamiltonian path in Kp where the multiset of edge lengths is S. Part of the motivation behind studying this particular problem is the possible application to constructing cyclic combinatorial designs. In this thesis we completely solve the case where S contains at most two distinct values.

#### Recommended Citation

Janiszewski, Susan, "An Exploration Into Two Combinatorial Problems" (2008). *Graduate College Dissertations and Theses*. 116.

http://scholarworks.uvm.edu/graddis/116