#### Abstract

If A is a square-free subset of an abelian group G, then the addition graph of A on G is the graph with vertex set G and distinct vertices x and y forming an edge if and only if x+y is in A. We prove that every connected cubic addition graph on an abelian group G whose order is divisible by 8 is Hamiltonian as well as every connected bipartite cubic addition graph on an abelian group G whose order is divisible by 4. We show that connected bipartite addition graphs are Cayley graphs and prove that every connected cubic Cayley graph on a group of dihedral type whose order is divisible by 4 is Hamiltonian.

#### Author Bio

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 honorssociety. 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.

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.

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.

#### Recommended Citation

Cheyne, Brian; Gupta, Vishal; and Wheeler, Coral
(2003)
"Hamilton Cycles in Addition Graphs,"
*Rose-Hulman Undergraduate Mathematics Journal*: Vol. 4
:
Iss.
1
, Article 6.

Available at:
http://scholar.rose-hulman.edu/rhumj/vol4/iss1/6

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