6 research outputs found
Small cycles, generalized prisms and Hamiltonian cycles in the Bubble-sort graph
The Bubble-sort graph , is a Cayley graph over the
symmetric group generated by transpositions from the set . It is a bipartite graph containing all even cycles of
length , where . We give an explicit
combinatorial characterization of all its - and -cycles. Based on this
characterization, we define generalized prisms in , and
present a new approach to construct a Hamiltonian cycle based on these
generalized prisms.Comment: 13 pages, 7 figure
Hamilton paths and cycles in vertex-transitive graphs of order 6p
It is shown that every connected vertex-transitive graph of order â–«â–«, where â–«â–« is a prime, contains a Hamilton path. Moreover, it is shown that, except for the truncation of the Petersen graph, every connected vertex-transitive graph of order â–«â–« which is not genuinely imprimitive contains a Hamilton cycle
Hamilton decompositions of 6-regular abelian Cayley graphs
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, Δi = Cay(A/(si), {sj1 , sj2}) is 4-regular, and Δi ≄ 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