Cartesian product of hypergraphs: properties and algorithms

Cartesian products of graphs have been studied extensively since the 1960s. They make it possible to decrease the algorithmic complexity of problems by using the factorization of the product. Hypergraphs were introduced as a generalization of graphs and the definition of Cartesian products extends naturally to them. In this paper, we give new properties and algorithms concerning coloring aspects of Cartesian products of hypergraphs. We also extend a classical prime factorization algorithm initially designed for graphs to connected conformal hypergraphs using 2-sections of hypergraphs

Square Property, Equitable Partitions, and Product-like Graphs

Equivalence relations on the edge set of a graph $G$ that satisfy restrictive conditions on chordless squares play a crucial role in the theory of Cartesian graph products and graph bundles. We show here that such relations in a natural way induce equitable partitions on the vertex set of $G$, which in turn give rise to quotient graphs that can have a rich product structure even if $G$ itself is prime.Comment: 20 pages, 6 figure

Nilpotent operators and weighted projective lines

We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those. The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type (2,3,p), obtained by Serre construction from the triangle singularity x^2+y^3+z^p. For p=6 the Ringel-Schmidmeier classification is thus covered by the classification of vector bundles for tubular type (2,3,6), and then is closely related to Atiyah's classification of vector bundles on a smooth elliptic curve. Returning to the general case, we establish that the stable categories associated to vector bundles or invariant subspaces of nilpotent operators may be naturally identified as triangulated categories. They satisfy Serre duality and also have tilting objects whose endomorphism rings play a role in singularity theory. In fact, we thus obtain a whole sequence of triangulated (fractional) Calabi-Yau categories, indexed by p, which naturally form an ADE-chain.Comment: More details added. 33 page

Opening Mirror Symmetry on the Quintic

Aided by mirror symmetry, we determine the number of holomorphic disks ending on the real Lagrangian in the quintic threefold. The tension of the domainwall between the two vacua on the brane, which is the generating function for the open Gromov-Witten invariants, satisfies a certain extension of the Picard-Fuchs differential equation governing periods of the mirror quintic. We verify consistency of the monodromies under analytic continuation of the superpotential over the entire moduli space. We reproduce the first few instanton numbers by a localization computation directly in the A-model, and check Ooguri-Vafa integrality. This is the first exact result on open string mirror symmetry for a compact Calabi-Yau manifold.Comment: 26 pages. v2: minor corrections and improvement