Refine
Document Type
- Preprint (8) (remove)
Keywords
- Kombinatorik (8) (remove)
Institute
- Mathematik (8)
- Informatik (1)
-
Counting polyominoes with minimum perimeter
(2005)
- Es wird die Anzahl der wesentlich verschiedenen Polyominoes der Ordnung n mit minimalem Umfang p(n) bestimmt.
-
On the characteristic of integral point sets in $\mathbb{E}^m$
(2005)
- We generalise the definition of the characteristic of an integral triangle to integral simplices and prove that each simplex in an integral point set has the same characteristic. This theorem is used for an efficient construction algorithm for integral point sets. Using this algorithm we are able to provide new exact values for the minimum diameter of integral point sets.
-
A note on Erdös-Diophantine graphs and Diophantine carpets
(2005)
- A Diophantine figure is a set of points on the integer grid $\mathbb{Z}^{2}$ where all mutual Euclidean distances are integers. We also speak of Diophantine graphs. The vertices are points in $\mathbb{Z}^{2}$ (the coordinates)and the edges are labeled with the distance between the two adjacent vertices, which is integral. In this language a Diophantine figure is a complete Diophantine graph. Two Diophantine graphs are equivalent if they only differ by translation or rotation of vertices. Due to a famous theorem of Erdös and Anning there are complete Diophantine graphs which are not contained in larger ones. We call them Erdös-Diophantine graphs. A special class of Diophantine graphs are Diophantine carpets. These are planar triangulations of a subset of the integer grid. We give an effective construction for Erdös-Diophantine graphs and characterize the chromatic number of Diophantine carpets.
-
Maximal integral point sets over Z^2
(2008)
- Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set P={p(1),...,p(n)} in Z^2 a maximal integral point set over Z^2 if all pairwise distances are integral and every additional point p(n+1) destroys this property. Here we consider such sets for a given cardinality and with minimum possible diameter. We determine some exact values via exhaustive search and give several constructions for arbitrary cardinalities. Since we cannot guarantee the maximality in these cases we describe an algorithm to prove or disprove the maximality of a given integral point set. We additionally consider restrictions as no three points on a line and no four points on a circle.
-
Enumeration of generalized polyominoes
(2006)
- Wir verallgemeinern den Begriff von Polyominoes (Tetrisbausteine) und betrachten Seite-an-Seite benachbarte überschneidungsfreie Vereinigungen von regelmäßigen k-Ecken. Für n<=4 geben wir Formeln für die Anzahl a_k(n) von verallgemeinerten Polyominoes, bestehend aus n regelmäßigen k-Ecken, an. Für weitere kleine Werte von k und n tabellieren wir durch computerunterstützte Enumeration gewonnene Anzahlen. Zum Abschluss erwähnen wir ein paar ungelöste Probleme für verallgemeinerte Polyominoes.
-
Inclusion-maximal integral point sets over finite fields
(2007)
- We consider integral point sets in affine planes over finite fields. Here an integral point set is a set of points in $GF(q)^2$ where the formally defined Euclidean distance of every pair of points is an element of $GF(q)$. From another point of view we consider point sets over $GF(q)^2$ with few and prescribed directions. So this is related to Redeis work. Another motivation comes from the field of ordinary integral point sets in Euclidean spaces. In this article we study the spectrum of integral point sets over $GF(q)^2$ which are maximal with respect to inclusion. We give some theoretical results, constructions, conjectures, and some numerical data.
-
On the minimum diameter of plane integral point sets
(2007)
- Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets P, which are sets of n points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter d(2,n) of a plane integral point set consisting of n points arises. We give some new exact values and describe state-of-the-art algorithms to obtain them. It turns out that plane integral point sets with minimum diameter consist very likely of subsets with many collinear points. For this special kind of point sets we prove a lower bound for d(2,n) achieving the known upper bound n^{c_2loglog n} up to a constant in the exponent.
-
Integral point sets over Z_n^m
(2007)
- There are many papers studying properties of point sets in the Euclidean space or on integer grids, with pairwise integral or rational distances. In this article we consider the distances or coordinates of the point sets which instead of being integers are elements of Z_n, and study the properties of the resulting combinatorial structures.
