Thoughts on Barnette's conjecture

We prove a new sufficient condition for a cubic 3-connected planar graph to be Hamiltonian. This condition is most easily described as a property of the dual graph. Let G be a planar triangulation. Then the dual G∗ is a cubic 3-connected planar graph, and G∗ is bipartite if and only if G is Eulerian. We prove that if the vertices of G are (improperly) coloured blue and red, such that the blue vertices cover the faces of G, there is no blue cycle, and every red cycle contains a vertex of degree at most 4, then G∗ is Hamiltonian. This result implies the following special case of Barnette’s Conjec- ture: if G is an Eulerian planar triangulation, whose vertices are properly coloured blue, red and green, such that every red-green cycle contains a vertex of degree 4, then G∗ is Hamiltonian. Our final result highlights the limitations of using a proper colouring of G as a starting point for proving Barnette’s Conjecture. We also explain related results on Bar- nette’s Conjecture that were obtained by Kelmans and for which detailed self-contained proofs have not been published

Neighborly and almost neighborly configurations, and their duals

This thesis presents new applications of Gale duality to the study of polytopes with extremal combinatorial properties. It consists in two parts. The first one is devoted to the construction of neighborly polytopes and oriented matroids. The second part concerns the degree of point configurations, a combinatorial invariant closely related to neighborliness. A d-dimensional polytope P is called neighborly if every subset of at most d/2 vertices of P forms a face. In 1982, Ido Shemer presented a technique to construct neighborly polytopes, which he named the "Sewing construction". With it he could prove that the number of neighborly polytopes in dimension d with n vertices grows superexponentially with n. One of the contributions of this thesis is the analysis of the sewing construction from the point of view of lexicographic extensions. This allows us to present a technique that we call the "Extended Sewing construction", that generalizes it in several aspects and simplifies its proof. We also present a second generalization that we call the "Gale Sewing construction". This construction exploits Gale duality an is based on lexicographic extensions of the duals of neighborly polytopes and oriented matroids. Thanks to this technique we obtain one of the main results of this thesis: a lower bound of ((r+d)^(((r+d)/2)^2)/(r^((r/2)^2)d^((d/2)^2)e^(3rd/4)) for the number of combinatorial types of neighborly polytopes of even dimension d and r+d+1 vertices. This result not only improves Shemer's bound, but it also improves the current best bounds for the number of polytopes. The combination of both new techniques also allows us to construct many non-realizable neighborly oriented matroids. The degree of a point configuration is the maximal codimension of its interior faces. In particular, a simplicial polytope is neighborly if and only if the degree of its set of vertices is [(d+1)/2]. For this reason, d-dimensional configurations of degree k are also known as "(d-k)-almost neighborly". The second part of the thesis presents various results on the combinatorial structure of point configurations whose degree is small compared to their dimension; specifically, those whose degree is smaller than [(d+1)/2], the degree of neighborly polytopes. The study of this problem comes motivated by Ehrhart theory, where a notion equivalent to the degree - for lattice polytopes - has been widely studied during the last years. In addition, the study of the degree is also related to the "generalized lower bound theorem" for simplicial polytopes, with Cayley polytopes and with Tverberg theory. Among other results, we present a complete combinatorial classification for point configurations of degree 1. Moreover, we show combinatorial restrictions in terms of the novel concept of "weak Cayley configuration" for configurations whose degree is smaller than a third of the dimension. We also introduce the notion of "codegree decomposition" and conjecture that any configuration whose degree is smaller than half the dimension admits a non-trivial codegree decomposition. For this conjecture, we show various motivations and we prove some particular cases

Refinements in Boundary Complexes of Polytopes

A complex °K is said to be a refinement of a complex L if there exists a homeomorphism Ψ Set K set L such that for each face L of L Ψ-1(L) is a union of faces of K. A face K of K is said to be principal if Ψ(K) is a face of L. Some results concerning 3-polytopes are shown not to extend to higher dimensions. For ≥ 4, there exist simple d-pclytopes with d+8 facets whose boundary complex cannot be expressed as a refinement of the boundary complex of v. d-polytope with d+7 facets. For ≥ 4, there exist simple d-polytopes whose graphs do not con¬tain refinements of the complete graph on d+1 vertices, if three particular vertices are preassigned as principal. A conjecture of Grunbaum is answered in the negative by constructing, for ≥ 4 simple d-polytopes P with d+4 facets in which two particular vertices may not be preassigned as principal if the boundary complex of P is expressed as a refinement of the boundary complex of a d-simplex; for d≥6, non-simple d-polytopes with d+3 facets having the same property are constructed. The main positive result is that the boundary complex of a d-polytope with d+2 facets, (d+3 facets if d = 4,5), may be ex¬pressed as a refinement of the boundary complex of the d-simplex with any two preassigned vertices principal. Several conjectures are made, among them the following genera¬lization of Balinsky's theorem on the d-connectedness of the graph of a d-polytope. If di+...+dk = ds di E ε N, then between any two vertices of ad-polytepe exist strong chains of di-faces, I = 1,...,k, disjoint except for the chosen vertices

Corruption of Religion and the Establishment Clause

Government neutrality toward religion is based on familiar considerations: the importance of avoiding religious conflict, alienation of religious minorities, and the danger that religious considerations will introduce a dangerous irrational dogmatism into politics and make democratic compromise more difficult. This paper explores one consideration, prominent at the time of the framing, that is often overlooked: the idea that religion can be corrupted by state involvement with it. This idea is friendly to religion but, precisely for that reason, is determined to keep the state away from religion. If the religion-protective argument for disestablishment is to be useful today, it cannot be adopted in the form in which it was understood in the 17th and 18th centuries, because in that form it is loaded with assumptions rooted in a particular variety of Protestant Christianity. Nonetheless, suitably revised, it provides a powerful reason for government, as a general matter, to keep its hands off religious doctrine. It offers the best explanation for many otherwise mysterious rules of Establishment Clause law

"Lights, Camera, History": Media Culture and the Kent State Shootings

Histor

Corruption of Religion and the Establishment Clause

Government neutrality toward religion is based on familiar considerations: the importance of avoiding religious conflict, alienation of religious minorities, and the danger that religious considerations will introduce a dangerous irrational dogmatism into politics and make democratic compromise more difficult. This Article explores one consideration, prominent at the time of the framing, that is often overlooked: the idea that religion can be corrupted by state involvement with it. This idea is friendly to religion but, precisely for that reason, is determined to keep the state away from religion. If the religion-protective argument for disestablishment is to be useful today, it cannot be adopted in the form in which it was understood in the seventeenth and eighteenth centuries, because in that form it is loaded with assumptions rooted in a particular variety of Protestant Christianity. Nonetheless, suitably revised, it provides a powerful reason for government, as a general matter, to keep its hands off religious doctrine. It offers the best explanation for many otherwise mysterious rules of Establishment Clause law

The social ethics of the Baptist Union of Southern Africa.

Abstract available in pdf file