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Minimum diameters of plane integral point sets
"... ABSTRACT. 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 ..."
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Cited by 12 (11 self)
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ABSTRACT. 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 stateoftheart 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 nc2 log log n up to a constant in the exponent. A famous question of Erdős asks for plane integral point sets with no 3 points on a line and no 4 points on a circle. Here, we talk of point sets in general position and denote the corresponding minimum diameter by ˙ d(2, n). Recently ˙ d(2, 7) = 22 270 could be determined via an exhaustive search. 1.
Small latin squares, quasigroups and loops
 Journal of Combinatorial Designs
, 2007
"... We present the numbers of isotopy classes and main classes of Latin squares, and the numbers of isomorphism classes of quasigroups and loops, up to order 10. The best previous results were for Latin squares of order 8 (Kolesova, Lam and Thiel, 1990), quasigroups of order 6 (Bower, 2000) and loops of ..."
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Cited by 11 (4 self)
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We present the numbers of isotopy classes and main classes of Latin squares, and the numbers of isomorphism classes of quasigroups and loops, up to order 10. The best previous results were for Latin squares of order 8 (Kolesova, Lam and Thiel, 1990), quasigroups of order 6 (Bower, 2000) and loops of order 7 (Brant and Mullen, 1985). The loops of order 8 have been independently found by \QSCGZ" and Guerin (unpublished, 2001). We also report on the most extensive search so far for a triple of mutually orthogonal Latin squares (MOLS) of order 10. Our computations show that any such triple must have only squares with trivial symmetry groups. 1
Minimum Sum and Difference Covers of Abelian Groups
, 2004
"... A subset S of a finite Abelian group G is said to be a sum cover of G if every element of G can be expressed as the sum of two not necessarily distinct elements in S, a strict sum cover of G if every element of G can be expressed as the sum of two distinct elements in S, and a difference cover of G ..."
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Cited by 6 (0 self)
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A subset S of a finite Abelian group G is said to be a sum cover of G if every element of G can be expressed as the sum of two not necessarily distinct elements in S, a strict sum cover of G if every element of G can be expressed as the sum of two distinct elements in S, and a difference cover of G if every element of G can be expressed as the difference of two elements in S. For each type of cover, we determine for small k the largest Abelian group for which a kelement cover exists. For this purpose we compute a minimum sum cover, a minimum strict sum cover, and a minimum difference cover for Abelian groups of order up to 85, 90, and 127, respectively, by a backtrack search with isomorph rejection.
On the characteristic of integral point sets in
 E m , Australas. J. Combin
"... 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 ..."
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Cited by 3 (3 self)
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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. Key words: integral distances, minimum diameter 2000 MSC: 52C10*, 11D99, 53C65 1
Computers and Discovery in Algebraic Graph Theory
 Edinburgh, 2001), Linear Algebra Appl
, 2001
"... We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory. ..."
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Cited by 1 (0 self)
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We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.
The Complete Catalog of 3Regular, Diameter3 Planar Graphs
, 1996
"... The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 ..."
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Cited by 1 (1 self)
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The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 vertices. An Upper Bound A graph with maximum degree \Delta and diameter D is called a (\Delta; D)graph. It is easily seen ([9], p. 171) that the order of a (\Delta,D)graph is bounded above by the Moore bound, which is given by 1+ \Delta + \Delta (\Delta \Gamma 1) + \Delta \Delta \Delta + \Delta(\Delta \Gamma 1) D\Gamma1 = 8 ? ! ? : \Delta(\Delta \Gamma 1) D \Gamma 2 \Delta \Gamma 2 if \Delta 6= 2; 2D + 1 if \Delta = 2: Figure 1: The regular (3,3)graph on 20 vertices (it is unique up to isomorphism) . For D 2 and \Delta 3, this bound is attained only if D = 2 and \Delta = 3; 7, and (perhaps) 57 [3, 14, 23]. Now, except for the case of C 4 (the cycle on four vertices), the num...
Enumeration of 2(9,3,λ) Designs and Their Resolutions
"... We consider 2(9,3,λ) designs, which are known to exist for all λ ≥ 1, and enumerate such designs for λ = 5 and their resolutions for 3 5, the smallest open cases. The number of nonisomorphic such structures obtained is 5 862 121 434, 426, 149 041, and 203 047 732, respectively. The designs are obta ..."
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We consider 2(9,3,λ) designs, which are known to exist for all λ ≥ 1, and enumerate such designs for λ = 5 and their resolutions for 3 5, the smallest open cases. The number of nonisomorphic such structures obtained is 5 862 121 434, 426, 149 041, and 203 047 732, respectively. The designs are obtained by an orderly algorithm, and the resolutions by two approaches: either by starting from the enumerated designs and applying a cliquefinding algorithm on two levels or by an orderly algorithm.
Unified Generation of Conformations, Conformers, and Stereoisomers: A Discrete MathematicsBased Approach
, 2008
"... In this feasibility study we propose an approach to a unified generation of stereoisomers including conformers of a molecular structure. The method is based on discrete mathematics, it recognizes and varies the orientation of in principle each quadruple of atoms. The method has potential to also gen ..."
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In this feasibility study we propose an approach to a unified generation of stereoisomers including conformers of a molecular structure. The method is based on discrete mathematics, it recognizes and varies the orientation of in principle each quadruple of atoms. The method has potential to also generate stereoisomers that are not describable in terms of stereocenters or single bond rotations. Fundamentals such as the concept of a (partial) orientation function are discussed, and mathematical tools such as Radon partitions and binary GrassmannPlücker relations are used to construct tests for realizability of abstract orientation functions. Simple examples are treated in detail. 1