Eulerian digraphs and toric Calabi-Yau varieties

We investigate the structure of a simple class of affine toric Calabi-Yau varieties that are defined from quiver representations based on finite eulerian directed graphs (digraphs). The vanishing first Chern class of these varieties just follows from the characterisation of eulerian digraphs as being connected with all vertices balanced. Some structure theory is used to show how any eulerian digraph can be generated by iterating combinations of just a few canonical graph-theoretic moves. We describe the effect of each of these moves on the lattice polytopes which encode the toric Calabi-Yau varieties and illustrate the construction in several examples. We comment on physical applications of the construction in the context of moduli spaces for superconformal gauged linear sigma models.Comment: 27 pages, 8 figure

Transitive Packing: A Unifying Concept in Combinatorial Optimization

This paper attempts to give a better understanding of the facial structure of previously separately investigated polyhedra. It introduces the notion of transitive packing and the transitive packing polytope. Polytopes that turn out to be special cases of the transitive packing polytope are, among others, the node packing polytope, the acyclic subdigraph polytope, the bipartite subgraph polytope, the planar subgraph polytope, the clique partitioning polytope, the partition polytope, the transitive acyclic subdigraph polytope, the interval order polytope, and the relatively transitive subgraph polytope. We give cutting plane proofs for several rich classes of valid inequalities of the transitive packing polytope,in this way introducing generalized cycle, generalized clique, generalized antihole, generalized antiweb, and odd partition inequalities. These classes subsume several known classes of valid inequalities for several of the special cases and give also many new inequalities for several other special cases. For some of the classes we also prove a lower bound for their Gomory-Chvdtal rank. Finally, we relate the concept of transitive packing to generalized (set) packing and covering as well as to balanced and ideal matrices