Title

Hamilton decompositions of 6-regular abelian Cayley graphs

Author

Erik E. Westlund, Michigan Technological University

Date of Award

2010

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

College, School or Department Name

Department of Mathematical Sciences

Advisor

Donald L Kreher

Abstract

In 1969, Lovasz asked whether every connected, vertex-transitive graph has a Hamilton path. This question has generated a considerable amount of interest, yet remains vastly open. To date, there exist no known connected, vertex-transitive graph that does not possess a Hamilton path. For the Cayley graphs, a subclass of vertex-transitive graphs, the following conjecture was made:

Weak Lovász Conjecture: Every nontrivial, finite, connected Cayley graph is hamiltonian.

The Chen-Quimpo Theorem proves that Cayley graphs on abelian groups flourish with Hamilton cycles, thus prompting Alspach to make the following conjecture:

Alspach Conjecture: Every 2k-regular, connected Cayley graph on a finite abelian group has a Hamilton decomposition.

Alspach’s conjecture is true for k = 1 and 2, but even the case k = 3 is still open. It is this case that this thesis addresses.

Chapters 1–3 give introductory material and past work on the conjecture. Chapter 3 investigates the relationship between 6-regular Cayley graphs and associated quotient graphs. A proof of Alspach’s conjecture is given for the odd order case when k = 3. Chapter 4 provides a proof of the conjecture for even order graphs with 3-element connection sets that have an element generating a subgroup of index 2, and having a linear dependency among the other generators.

Chapter 5 shows that if Γ = Cay(A, {s₁, s₂, s₃}) is a connected, 6-regular, abelian Cayley graph of even order, and for some1 ≤ i ≤ 3, Δ_i = Cay(A/(s_i), {sj1 , sj2}) is 4-regular, and Δ_i ≄ Cay(ℤ₃, {1, 1}), then Γ has a Hamilton decomposition. Alternatively stated, if Γ = Cay(A, S) is a connected, 6-regular, abelian Cayley graph of even order, then Γ has a Hamilton decomposition if S has no involutions, and for some s ∈ S, Cay(A/(s), S) is 4-regular, and of order at least 4.

Finally, the Appendices give computational data resulting from C and MAGMA programs used to generate Hamilton decompositions of certain non-isomorphic Cayley graphs on low order abelian groups.

Recommended Citation

Westlund, Erik E., "Hamilton decompositions of 6-regular abelian Cayley graphs", Dissertation, Michigan Technological University, 2010.

https://doi.org/10.37099/mtu.dc.etds/206