Hamiltonian decompositions of Cayley graphs on abelian groups of even order

AbstractAlspach conjectured that any 2k-regular connected Cayley graph cay(A,S) on a finite abelian group A can be decomposed into k hamiltonian cycles. In 1992, the author proved that the conjecture holds if S={s1,s2,…,sk} is a minimal generating set of an abelian group A of odd order. Here we prove an analogous result for abelian group of even order: If A is a finite abelian group of even order at least 4 and S={s1,s2,…,sk} is a strongly minimal generating set (i.e., 2si∉〈S−{si}〉 for each 1⩽i⩽k) of A, then cay(A,S) can be decomposed into hamiltonian cycles

On Hamilton decompositions of infinite circulant graphs

The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph). Although it is known that every connected 2k-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k-valent connected circulant graph has a decomposition into k edge-disjoint Hamilton cycles. We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the connection set is ±{1,2,...,k} or ±{1,2,...,k - 1, 1,2,...,k + 1}

Hamilton Decompositions of Certain 6-regular Cayley Graphs on Abelian Groups with a Cyclic Subgroup of Index Two

Alspach conjectured that every connected Cayley graph of even valency on a finite Abelian group is Hamilton-decomposable. Using some techniques of Liu, this article shows that if A is an Abelian group of even order with a generating set {a,b}, and A contains a subgroup of index two, generated by c, then the 6-regular Cayley graph is Hamilton-decomposable

Automorphisms generating disjoint Hamilton cycles in star graphs

In the first part of the thesis we define an automorphism φn for each star graph Stn of degree n − 1, which yields permutations of labels for the edges of Stn taken from the set of integers {1, . . . , bn/2c}. By decomposing these permutations into permutation cycles, we are able to identify edge-disjoint Hamilton cycles that are automorphic images of a known two-labelled Hamilton cycle H1 2(n) in Stn. Our main result is an improvement from the existing lower bound of bϕ(n)/10c to b2ϕ(n)/9c, where ϕ is Euler’s totient function, for the known number of edge-disjoint Hamilton cycles in Stn for all odd integers n. For prime n, the improvement is from bn/8c to bn/5c. We extend this result to the cases when n is the power of a prime other than 3 and 7. The second part of the thesis studies ‘symmetric’ collections of edge-disjoint Hamilton cycles in Stn, i.e. collections that comprise images of H1 2(n) under general label-mapping automorphisms. We show that, for all even n, there exists a symmetric collection of bϕ(n)/2c edge-disjoint Hamilton cycles, and Stn cannot have symmetric collections of greater than bϕ(n)/2c such cycles for any n. Thus, Stn is not symmetrically Hamilton decomposable if n is not prime. We also give cases of even n, in terms of Carmichael’s reduced totient function λ, for which ‘strongly’ symmetric collections of edge-disjoint Hamilton cycles, which are generated from H1 2(n) by a single automorphism, can and cannot attain the optimum bound bϕ(n)/2c for symmetric collections. In particular, we show that if n is a power of 2, then Stn has a spanning subgraph with more than half of the edges of Stn, which is strongly symmetrically Hamilton decomposable. For odd n, it remains an open problem as to whether the bϕ(n)/2c can be achieved for symmetric collections, but we are able to show that, for certain odd n, a ϕ(n)/4 bound is achievable and optimal for strongly symmetric collections. The search for edge-disjoint Hamilton cycles in star graphs is important for the design of interconnection network topologies in computer science. All our results improve on the known bounds for numbers of any kind of edge-disjoint Hamilton cycles in star graphs

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