Homomorphisms of binary Cayley graphs

A binary Cayley graph is a Cayley graph based on a binary group. In 1982, Payan proved that any non-bipartite binary Cayley graph must contain a generalized Mycielski graph of an odd-cycle, implying that such a graph cannot have chromatic number 3. We strengthen this result first by proving that any non-bipartite binary Cayley graph must contain a projective cube as a subgraph. We further conjecture that any homo- morphism of a non-bipartite binary Cayley graph to a projective cube must be surjective and we prove some special case of this conjecture

Short proofs of some extremal results

We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.Comment: 19 page

Homology for higher-rank graphs and twisted C*-algebras

We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C*-algebra of a k-graph by a T-valued 2-cocycle and demonstrate that examples include all noncommutative tori. In the appendices, we construct a cubical set \tilde{Q}(\Lambda) from a k-graph {\Lambda} and demonstrate that the homology and topological realisation of {\Lambda} coincide with those of \tilde{Q}(\Lambda) as defined by Grandis.Comment: 33 pages, 9 pictures and one diagram prepared in TiK

Polygonal Complexes and Graphs for Crystallographic Groups

The paper surveys highlights of the ongoing program to classify discrete polyhedral structures in Euclidean 3-space by distinguished transitivity properties of their symmetry groups, focussing in particular on various aspects of the classification of regular polygonal complexes, chiral polyhedra, and more generally, two-orbit polyhedra.Comment: 21 pages; In: Symmetry and Rigidity, (eds. R.Connelly, A.Ivic Weiss and W.Whiteley), Fields Institute Communications, to appea

Problems on Polytopes, Their Groups, and Realizations

The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete Geometry, to appear