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.

Hamilton-Waterloo problem with triangle and C9 factors

The Hamilton-Waterloo problem and its spouse-avoiding variant for uniform cycle sizes asks if Kv, where v is odd (or Kv - F, if v is even), can be decomposed into 2-factors in which each factor is made either entirely of m-cycles or entirely of n-cycles. This thesis examines the case in which r of the factors are made up of cycles of length 3 and s of the factors are made up of cycles of length 9, for any r and s. We also discuss a constructive solution to the general (m,n) case which fixes r and s.

Transatlantic Echoes. Alexander von Humboldt in World Literature

Alexander von Humboldt (1769–1859) was a world traveler, bestselling writer, and versatile researcher, a European salon sensation, and global celebrity. Yet the enormous literary echo he generated has remained largely unexplored. Humboldt inspired generations of authors, from Goethe and Byron to Enzensberger and García Márquez, to reflect on cultural difference, colonial ideology, and the relation between aesthetics and science. This collection of one-hundred texts features tales of adventure, travel reports, novellas, memoirs, letters, poetry, drama, screenplays, and even comics—many for the first time in English. The selection covers the foundational myths and magical realism of Latin America, the intellectual independence of Emerson, Thoreau, Poe, and Whitman in the United States, discourses in Imperial, Weimar, Nazi, East, and West Germany, as well as recent films and fiction. This documented source book addresses scholars in cultural and postcolonial studies as well as readers in history and comparative literature.

Cosmos and Colonialism. Alexander von Humboldt in Cultural Criticism

Alexander von Humboldt explored the Spanish Empire on the verge of its collapse (1799–1804). He is the most significant German travel writer and the most important mediator between Europe and the Americas of the nineteenth century. His works integrated knowledge from two dozen domains. Today, he is at the center of debates on imperial discourse, postcolonialism, and globalization. This collection of fifty essays brings together a range of responses, many presented here for the first time in English. Authors from Schiller, Chateaubriand, Sarmiento, and Nietzsche, to Robert Musil, Kurt Tucholsky, Ernst Bloch, and Alejo Carpentier paint the historical background. Essays by contemporary travel writers and recent critics outline the current controversies on Humboldt. The source materials collected here will be indispensable to scholars of German, French, and Latin and North American literature as well as cultural and postcolonial studies, history, art history, and the history of science.