We consider cellular automata on Cayley graphs and compare their computational powers according to the architecture on which t h e y work. We show that, if there exists a homomorphism with a nite kernel from a group into another one such that the image of the rst group has a nite index in the second one, then every cellular automaton on the Cayley graph of one of these groups can be uni-formall...

A hypergraph is b-simple if no two distinct edges share more than b vertices. Let m(r, t, g) denote the minimum number of edges in an r-uniform non-t-colorable hypergraph of girth at least g. Erdős and Lovász proved that m(r, t, 3) ≥ t 2(r−2) 16r(r − 1)2 and m(r, t, g) ≤ 4 · 20g−1r3g−5t(g−1)(r+1). A result of Szabó improves the lower bound by a factor of r2− for sufficiently large r. We improve...

In 1987 Paul Erdős asked me if the Cayley graph defined on Z by a lacunary sequence has necessarily a finite chromatic number. Below is my answer, delivered to him on the spot but neverpublished, and some additional remarks. The key is the interpretation of the question in terms of return times of dynamical systems. 1 The Cayley graph ZΛ 1.1 Let Λ ⊂ N. By definition, the Cayley graph ZΛ is the ...

In line with symmetrical graphs such as Cayley graphs and vertex transitive graphs, we introduce a new class of symmetrical graphs called diametrically uniform graphs. The class of diametrically uniform graphs includes vertex transitive graphs and hence Cayley graphs. A tree t-spanner of graph G is a spanning tree T in which the distance between every pair of vertices is at most t times their d...

Fix r ≥ 2 and a collection of r-uniform hypergraphs H. What is the minimum number of edges in an H-free r-uniform hypergraph with chromatic number greater than k? We investigate this question for various H. Our results include the following: • An (r, l)-system is an r-uniform hypergraph with every two edges sharing at most l vertices. For k sufficiently large, there is an (r, l)-system with chr...

When working with diagrams in visual environments like graphical diagram editors, diagrams have to be represented by an internal model. Graphs and hypergraphs are well-known concepts for such internal models. This paper shows how hypergraphs can be uniformly used for a wide range of different diagram types where hyperedges are used to represent diagram components as well as spatial relationship...

Sketching, i.e., drawing diagrams by hand and directly on the screen, is gaining popularity, as it is a comfortable and natural way to create and edit diagrams. Hand drawing is inherently imprecise, and often sloppy. As a consequence, when processing hand drawn diagrams with a computer, ambiguities arise: it is not always clear what part of the drawing is meant to represent what component. Reso...

DiaGen is a specification method, which is primarily based on a hypergraph grammar, and a tool that allows to automatically generate diagram editors from such a specification. Generated editors are free-hand editors, but with an automatic, constraint-based layout for correct diagrams. A hypergraph parser checks diagram correctness and makes it possible to translate diagrams into some user-defin...

A partial Steiner (k, l)-system is a k-uniform hypergraph G with the property that every l-element subset of V is contained in at most one edge of G . In this paper we show that for given k, l and t there exists a partial Steiner (k, l)-system such that whenever an l-element subset from every edge is chosen, the resulting l-uniform hypergraph contains a clique of size t. As the main result of t...

Learning a hidden hypergraph is a natural generalization of the classical group testing problem that consists in detecting unknown hypergraph Hun = H(V, E) by carrying out edge-detecting tests. In the given paper we focus our attention only on a specific family F(t, s, `) of localized hypergraphs for which the total number of vertices |V | = t, the number of edges |E| 6 s, s ¿ t, and the cardin...