We analyse exhaustively the structure of non-degenerate Cauchy horizons in Gowdy space-times, and we establish existence of a large class of non-polarized Gowdy space-times with such horizons. Added in proof: Our results here, together with deep new results of H. Ringström (talk at the Miami Waves conference, January 2004), establish strong cosmic censorship in (toroidal) Gowdy space-times.

In this paper, we construct gauge bundles on a noncommutative toroidal orbifold Tθ/Z2. First, we explicitly construct a bundle with constant curvature connections on a noncommutative Tθ following Rieffel’s method. Then, applying the appropriate quotient conditions for its Z2 orbifold, we find a Connes-Douglas-Schwarz type solution of matrix theory compactified on Tθ/Z2. When we consider two cop...

We prove two results relating 3-manifold groups to fundamental groups occurring in complex geometry. Let N be a compact, connected, orientable 3-manifold. If N has non-empty, toroidal boundary, and π1(N) is a Kähler group, then N is the product of a torus with an interval. On the other hand, if N has either empty or toroidal boundary, and π1(N) is a quasi-projective group, then all the prime co...

For each positive integer n, the toroidal queens graph may be described as a graph with vertex set Zn × Zn where every vertex is adjacent to those vertices in the directions (1, 0), (0, 1), (1, 1), (1,−1) from it. We here extend this idea, examining graphs with vertex set F × F , where F is a finite field, and any four directions are used to define adjacency. The automorphism groups and isomorp...

We consider the degree/diameter problem for graphs embedded in a surface, namely, given a surface Σ and integers ∆ and k, determine the maximum order N(∆, k,Σ) of a graph embeddable in Σ with maximum degree ∆ and diameter k. We introduce a number of constructions which produce many new largest known planar and toroidal graphs. We record all these graphs in the available tables of largest known ...

We provide a description of unlabelled enumeration techniques, with complete proofs, for graphs that can be canonically obtained by substituting 2-pole networks for the edges of core graphs. Using structure theorems for toroidal and projectiveplanar graphs containing no K3,3-subdivisions, we apply these techniques to obtain their unlabelled enumeration.

Two embeddings Ψ1 and Ψ2 of a graph G in a surface Σ are equivalent if there is a homeomorphism of Σ to itself carrying Ψ1 to Ψ2. In this paper, we classify the flexibility of embeddings in the torus with representativity at least 4. We show that if a graph G has an embedding Ψ in the torus with representativity at least 4, then one of the following holds: (i) Ψ is the unique embedding of G in ...

If X is a collection of edges in a graph G, let G/X denote the contraction of X. Following a question of Oxley and a conjecture of Oporowski, we prove that every projective-planar graph G admits an edge-partition {X,Y } such that G/X and G/Y have tree-width at most three. We prove that every toroidal graph G admits an edge-partition {X,Y } such that G/X and G/Y have tree-width at most three and...