Dear Madam/Sir,
I am Yury J. Ionin, Professor of the Department of Mathematics at Central Michigan University, Mt. Pleasant, MI 48859. I am submitting the paper "Hamilton cycles in addition graphs" for publication in RHIT Mathematics Journal.
The paper was prepared under my supervision by undergraduate students Brian Cheyne (Western Michigan University), Vishal Gupta (Yale University), and Coral Wheeler (University of Akron) during the summer program in June-July 2002 at Central Michigan University. The program was funded by the REU NSF Grant DMS-0097394.
The paper introduces a new type of regular graphs, addition graphs, related to finite abelian groups.
The goal of the research carried by the students was to determine which of these graphs are Hamiltonian.
The main results of the paper establish that connected cubic addition graphs over groups, whose order is
divisible by 8, are always Hamiltonian as well as bipartite connected cubic addition graphs over groups,
whose order is divisible by 4. Moreover, bipartite addition graphs over abelian groups are in fact
Cayley graphs over certain non-abelian groups. This observation led to obtaining a previously
unknown class of non-abelian groups, over which all connected cubic Cayley graphs are Hamiltonian.
These are the so-called groups of dihedral type, that is, the groups that have an abelian subgroup of index 2
such that all elements of the complement of this subgroup are of order 2.
The paper proves that all connected cubic Cayley graphs
over groups of dihedral type, whose order is divisible by 4, are Hamiltonian.
Professor Robert Molina, Alma College, Alma, MI, 48801 (molina@alma.edu) has agreed as the reference for the paper.
Below are the statements made by the students.
From Brian Cheyne (bdcheyne@excite.com)
Our research was conducted from May 27, 2002 until July 19, 2002,
at which time I was an undergraduate at Western Michigan University. I am
a junior at Western Michigan University majoring in General Mathematics and Economics. Last year I learned of the National Science Foundation's Research Experience for Undergraduates program through my involvement in Pi Mu Epsilon, a math honors society. Since I am planning on attending graduate school for either math or economics, I figured that would be the perfect opportunity for me to get my feet wet in research. Over the summer of 2002, I participated in this program at Central Michigan University, and was introduced to our problem by Dr. Yury Ionin. Under his supervision, Coral, Vishal, and I worked on our project. When I'm not working on homework, I enjoy contact sports and am involved in club wrestling and rugby.
From Vishal Gupta (vishal.gupta@yale.edu)
To Whom It May Concern,
I hereby request that the paper "Hamilton Cycles in Addition Graphs" on
which I am a co-author with Ms. Coral Wheeler and Mr. Brian Cheyne be published in the RHIT Mathematics Journal.
The research for this project was done between May 28th and July 19th, 2002 during which time I was an undergraduate at Yale University.
Please use the following paragraph in the online journal:
I am an undergraduate at Yale University expecting to graduate in May 2004
with a B.A. in Mathematics and Philosophy. This research was done as part
of a Research Experience for Undergraduates (REU) Program at Central Michigan University. I plan to continue researching Mathematics, pursue a
Ph.D. and ideally work in mathematical research outside academia.
Dr. Yury Ionin of Central Michigan University, was our mentor on this
project.
Sincerely,
Vishal Gupta
From Coral Wheeler (xxsunkistxx@hotmail.com)
I, Coral Wheeler, conducted research on Addition Graphs from May 27th to July 19th in 2002. I was an undergraduate at that time.
I am a mathematics and Physics major at the University of Akron. I plan on attending graduate school for one of the subjects or possibly a combination of both after I graduate. The work on Addition Graphs that I completed was done at the National Science Foundation Research Experience for Undergraduates at Central Michigan University. I also currently am involved in two research projects at my own University. One involves the Fibobacci sequence and the other the coherence length of white and monochromatic light. I also work part time as math tutor.