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.

Effects of Chemical Deicers on Amphibian Communites

Roads and highways present a unique challenge to wildlife as they exhibit substantial impacts on the surrounding ecosystem through the interruption of a number of ecological processes. With new roads added to the national highway system every year, an understanding of these impacts is required for effective mitigation of potential environmental impacts. A major contributor to these negative effects is the deposition of chemicals used in winter deicing activities to nearby surface waters. These chemicals often vary in composition and may affect freshwater species differently. The negative impacts of widespread deposition of sodium chloride (NaCl) have prompted a search for an `environmentally friendly' alternative. However, little research has investigated the potential environmental effects of widespread use of these alternatives. Herein, I detail the results of laboratory tests and field surveys designed to determine the impacts of road salt (NaCl) and other chemical deicers on amphibian communities in Michigan's Upper Peninsula. Using larval amphibians I demonstrate the lethal impacts of a suite of chemical deicers on this sensitive, freshwater species. Larval wood frogs (Lithobates sylvatica) were tolerant of short-term (96 hours) exposure to urea (CH4N2O), sodium chloride (NaCl), and magnesium chloride (MgCl2). However, these larvae were very sensitive to acetate products (C8H12CaMgO8, CH3COOK) and calcium chloride (CaCl2). These differences in tolerance suggest that certain deicers may be more harmful to amphibians than others. Secondly, I expanded this analysis to include an experiment designed to determine the sublethal effects of chronic exposure to environmentally realistic concentrations of NaCl on two unique amphibian species, L. sylvatica and green frogs (L. clamitans). L. sylvatica tend to breed in small, ephemeral wetlands and metamorphose within a single season. However, L. clamitans breed primarily in more permanent wetlands and often remain as tadpoles for one year or more. These species employ different life history strategies in this region which may influence their response to chronic NaCl exposure. Both species demonstrated potentially harmful effects on individual fitness. L. sylvatica larvae had a high incidence of edema suggesting the NaCl exposure was a significant physiologic stressor to these larvae. L. clamitans larvae reduced tail length during their exposure which may affect adult fitness of these individuals. In order to determine the risk local amphibians face when using these roadside pools, I conducted a survey of the spatial distribution of chloride in the three northernmost counties of Michigan. This area receives a relatively low amount of NaCl which is confined to state and federal highways. The chloride concentrations in this region were much lower than those in urban systems; however, amphibians breeding in the local area may encounter harmful chloride levels arising from temporal variations in hydroperiods. Spatial variation of chloride levels suggests the road-effect zone for amphibians may be as large as 1000 m from a salt-treated highway. Lastly, I performed an analysis of the use of specific conductance to predict chloride concentrations in natural surface water bodies. A number of studies have used this regression to predict chloride concentrations from measurements of specific conductance. This method is often chosen in the place of ion chromatography due to budget and time constraints. However, using a regression method to characterize this relationship does not result in accurate chloride ion concentration estimates.