FACTORIZING SMALL 2-GROUPS SÁNDOR SZABÓ Let G be a finite abelian group and let G = A 1 • • • A n... more FACTORIZING SMALL 2-GROUPS SÁNDOR SZABÓ Let G be a finite abelian group and let G = A 1 • • • A n be a factorization of G into its subsets A 1 ,. .. , A n. For a given G certain choices of the orders |A 1 |,. .. , |A n | guarantee that one of the factors is periodic. In connection with an open problem we determine such choices of orders of factors in two special cases. In these cases |G| is either 2 5 or 2 6 .
Egy adott gráfban a maximálisélsúlyú klikk megtalálása egy ismertés fontos probléma, sok alkalmaz... more Egy adott gráfban a maximálisélsúlyú klikk megtalálása egy ismertés fontos probléma, sok alkalmazással. A probléma megoldására léteznek a lineáris programozás eszközeit, valamint lineáris programozással nem kapcsolatos kombinatorikus alapú, keresési fát használó algoritmusok is. Egy olyan tanulmányhoz [6] fűzünk megjegyzéseket, amelyben a szerzők egy kombinatorikus alapú algoritmust hasonlítottakössze kettő lineáris programozás, es egy kvadratikus programozás alapú algoritmussal. Az [1]és [6] cikkben szereplő egyik programot módosítjuk. A módosításokhoz a gráf csúcsainaḱ eséleinek színezésével jutunk. Azúj programokat numerikus kísérletekben teszteltük.
In many clique search algorithms well coloring of the nodes is employed to find an upper bound of... more In many clique search algorithms well coloring of the nodes is employed to find an upper bound of the clique number of the given graph. In an earlier work a non-traditional edge coloring scheme was proposed to get upper bounds that are typically better than the one provided by the well coloring of the nodes. In this paper we will show that the same scheme for well coloring of the edges can be used to find lower bounds for the clique number of the given graph. In order to assess the performance of the procedure we carried out numerical experiments.
We consider a graph whose vertices are legally colored using k colors and ask if the graph contai... more We consider a graph whose vertices are legally colored using k colors and ask if the graph contains a k-clique. As it turns out this very special type of k-clique problem is in an intimate connection with constructing schedules. The practicality this clique search based construction of schedules is checked by carrying out numerical experiments.
It is known that the problem of proper coloring of the nodes of a given graph can be reduced to f... more It is known that the problem of proper coloring of the nodes of a given graph can be reduced to finding cliques in a suitably constructed auxiliary graph. In this work, we explore the possibility of reducing the search space by exploiting the symmetries present in the auxiliary graph. The proposed method can also be used for efficient exact coloring of hyper graphs. We also precondition the auxiliary graph in order to further reduce the search space. We carry out numerical experiments to assess the practicality of these proposals. We solve some hard cases and prove a new lower limit of seven for the mycielski7 graph with the aid of the proposed technique.
Proceedings of the American Mathematical Society, 1983
In [7] S. K. Stein constructed a 10-dimensional centrally-symmetric star body whose translates ti... more In [7] S. K. Stein constructed a 10-dimensional centrally-symmetric star body whose translates tile 10 10 -space but whose translates by a lattice do not tile it. In [8] he constructed a 5 5 -dimensional star polyhedron whose translates tile 5 5 -space but whose congruent copies by a group of motions do not tile it. So there is no lattice tiling by translates of this polyhedron. In the present paper we shall construct a 5 5 -dimensional centrally-symmetric star polyhedron whose translates tile 5 5 -space but whose congruent copies by a group of motions do not tile it. Furthermore, this phenomenon occurs at an infinitude of dimensions.
Proceedings of the American Mathematical Society, 1983
Consider the set of closed unit cubes whose edges are parallel to the coordinate unit vectors e 1... more Consider the set of closed unit cubes whose edges are parallel to the coordinate unit vectors e 1 , … , e n {{\mathbf {e}}_1}, \ldots ,{{\mathbf {e}}_n} and whose centers are i e j i{{\mathbf {e}}_j} , 0 ⩽ | i | ⩽ k 0 \leqslant |i| \leqslant k , in n n -dimensional Euclidean space. The union of these cubes is called a cross. This cross consists of 2 k n + 1 2kn + 1 cubes; a central cube together with 2 n 2n arms of length k k . A family of translates of a cross whose union is n n -dimensional Euclidean space and whose interiors are disjoint is a tiling. Denote the set of translation vectors by L {\mathbf {L}} . If the vector set L {\mathbf {L}} is a vector lattice, then we say that the tiling is a lattice tiling. If every vector of L {\mathbf {L}} has rational coordinates, then we say that the tiling is a rational tiling, and, similarly, if every vector of L {\mathbf {L}} has integer coordinates, then we say that the tiling is an integer tiling. Is there a noninteger tiling by cross...
In this paper, we single out the following particular case of the clique search problem. The vert... more In this paper, we single out the following particular case of the clique search problem. The vertices of the given graph are legally colored with k colors and we are looking for a clique with k nodes in the graph. In other words, we want to decide if a given k-partite graph contains a clique with k nodes. The maximum clique problem asks for finding a maximum clique in a given finite simple graph. The problem of deciding if the given graph contains a clique with k vertices is called the k-clique problem. The first problem is NP-hard and the second one is NP-complete. The special clique search problem, we propose, is still an NP-complete problem. We will show that the k-clique problem in the special case of k-partite graphs is more tractable than in the general case. In order to illustrate the possible practical utility of this restricted type clique search problem we will show that the job shop scheduling problem can be reduced to such a clique search problem in a suitable constructe...
In 1961 J. Stein proposed an algorithm to compute the greatest common divisor of two integers. In... more In 1961 J. Stein proposed an algorithm to compute the greatest common divisor of two integers. In this paper we point out that similar algorithms exist in the ring of integers of various quadratic number fields and also in the non-commutative ring of the Hurwitz quaternions. The implementations of the algorithms are straightforward. However the procedures vary from case to case. AMS Mathematics Subject Classification (2010): Primary 11A51, Secondary 11Y16 Key words and phrases: gcd in quadratic number fields, Hurwitz quater-nion, binary gcd algorithm and its extensions.
G. Hajós proved that if a finite abelian group is a direct product of its cyclic subsets, then at... more G. Hajós proved that if a finite abelian group is a direct product of its cyclic subsets, then at least one of the factors must be a subgroup. We give a new elementary proof of this theorem based on the special case for p-groups.
The paper will present a method to establish an upper bound on the clique number of a given finit... more The paper will present a method to establish an upper bound on the clique number of a given finite simple graph. In order to evaluate the performance of the proposed algorithm in practice we carry out a large scale numerical experiment on carefully selected benchmark instances. Povzetek: Razvita in opisana je nova metoda za določanje zgornje meje števila klik v grafu.
Three infinite families of finite abeliab groups will be described such that each members of thes... more Three infinite families of finite abeliab groups will be described such that each members of these families has the Redei k-property for many non-trivial values of k.
A spectral set in R^n is a set X of finite Lebesgue measure such that L^2(X) has an orthogonal ba... more A spectral set in R^n is a set X of finite Lebesgue measure such that L^2(X) has an orthogonal basis of exponentials. It is conjectured that every spectral set tiles R^n by translations. A set of translations T has a universal spectrum if every set that that tiles by translations by T has this spectrum. A recent result proved that many periodic tiling sets have universal spectra, using results from factorizations of abelian groups, for groups for which a strong form of a conjecture of Tijdeman is valid. This paper shows Tijdeman's conjecture does not hold for the cyclic group of order 900. It formulates a new sufficient conjecture for a periodic tiling set to have a universal spectrum, and uses it to show that the tiling sets for the counterexample above do have universal spectra.
By a theorem of L. Rédei if a finite abelian group is a direct product of its subsets such that e... more By a theorem of L. Rédei if a finite abelian group is a direct product of its subsets such that each subset has a prime number of elements and contains the identity element of the group, then at least one of the factors must be a subgroup. The content of this paper is that this result holds for certain infinite abelian groups, too. Namely, for groups that are direct products of finitely many Prüferian groups and finite cyclic groups of prime power order, belonging to pairwise distinct primes.
The paper deals with the following problem: If a finite abelian 2-group is a direct product of it... more The paper deals with the following problem: If a finite abelian 2-group is a direct product of its subsets of cardinality 4, does it follow that at least one of the factors is periodic? Two results are presented. In the first one, the structures of the group and the subsets are restricted but the size of the the group is not. In the second one, the group and the factors are general but the order of the group is 26.
If |A1|= q1, . . . , |An|= qn, then (q1, . . . ,qn) is called the type of the factorization A1 · ... more If |A1|= q1, . . . , |An|= qn, then (q1, . . . ,qn) is called the type of the factorization A1 · · ·An. By the fundamental theorem of finite abelian groups each finite abelian group is a direct product of cyclic groups. This decomposition into cyclic groups is not necessarily unique. However, if G is the direct product of cyclic groups of orders t1, . . . , tr respectively, then we say that G is of type (t1, . . . , tr). We use
FACTORIZING SMALL 2-GROUPS SÁNDOR SZABÓ Let G be a finite abelian group and let G = A 1 • • • A n... more FACTORIZING SMALL 2-GROUPS SÁNDOR SZABÓ Let G be a finite abelian group and let G = A 1 • • • A n be a factorization of G into its subsets A 1 ,. .. , A n. For a given G certain choices of the orders |A 1 |,. .. , |A n | guarantee that one of the factors is periodic. In connection with an open problem we determine such choices of orders of factors in two special cases. In these cases |G| is either 2 5 or 2 6 .
Egy adott gráfban a maximálisélsúlyú klikk megtalálása egy ismertés fontos probléma, sok alkalmaz... more Egy adott gráfban a maximálisélsúlyú klikk megtalálása egy ismertés fontos probléma, sok alkalmazással. A probléma megoldására léteznek a lineáris programozás eszközeit, valamint lineáris programozással nem kapcsolatos kombinatorikus alapú, keresési fát használó algoritmusok is. Egy olyan tanulmányhoz [6] fűzünk megjegyzéseket, amelyben a szerzők egy kombinatorikus alapú algoritmust hasonlítottakössze kettő lineáris programozás, es egy kvadratikus programozás alapú algoritmussal. Az [1]és [6] cikkben szereplő egyik programot módosítjuk. A módosításokhoz a gráf csúcsainaḱ eséleinek színezésével jutunk. Azúj programokat numerikus kísérletekben teszteltük.
In many clique search algorithms well coloring of the nodes is employed to find an upper bound of... more In many clique search algorithms well coloring of the nodes is employed to find an upper bound of the clique number of the given graph. In an earlier work a non-traditional edge coloring scheme was proposed to get upper bounds that are typically better than the one provided by the well coloring of the nodes. In this paper we will show that the same scheme for well coloring of the edges can be used to find lower bounds for the clique number of the given graph. In order to assess the performance of the procedure we carried out numerical experiments.
We consider a graph whose vertices are legally colored using k colors and ask if the graph contai... more We consider a graph whose vertices are legally colored using k colors and ask if the graph contains a k-clique. As it turns out this very special type of k-clique problem is in an intimate connection with constructing schedules. The practicality this clique search based construction of schedules is checked by carrying out numerical experiments.
It is known that the problem of proper coloring of the nodes of a given graph can be reduced to f... more It is known that the problem of proper coloring of the nodes of a given graph can be reduced to finding cliques in a suitably constructed auxiliary graph. In this work, we explore the possibility of reducing the search space by exploiting the symmetries present in the auxiliary graph. The proposed method can also be used for efficient exact coloring of hyper graphs. We also precondition the auxiliary graph in order to further reduce the search space. We carry out numerical experiments to assess the practicality of these proposals. We solve some hard cases and prove a new lower limit of seven for the mycielski7 graph with the aid of the proposed technique.
Proceedings of the American Mathematical Society, 1983
In [7] S. K. Stein constructed a 10-dimensional centrally-symmetric star body whose translates ti... more In [7] S. K. Stein constructed a 10-dimensional centrally-symmetric star body whose translates tile 10 10 -space but whose translates by a lattice do not tile it. In [8] he constructed a 5 5 -dimensional star polyhedron whose translates tile 5 5 -space but whose congruent copies by a group of motions do not tile it. So there is no lattice tiling by translates of this polyhedron. In the present paper we shall construct a 5 5 -dimensional centrally-symmetric star polyhedron whose translates tile 5 5 -space but whose congruent copies by a group of motions do not tile it. Furthermore, this phenomenon occurs at an infinitude of dimensions.
Proceedings of the American Mathematical Society, 1983
Consider the set of closed unit cubes whose edges are parallel to the coordinate unit vectors e 1... more Consider the set of closed unit cubes whose edges are parallel to the coordinate unit vectors e 1 , … , e n {{\mathbf {e}}_1}, \ldots ,{{\mathbf {e}}_n} and whose centers are i e j i{{\mathbf {e}}_j} , 0 ⩽ | i | ⩽ k 0 \leqslant |i| \leqslant k , in n n -dimensional Euclidean space. The union of these cubes is called a cross. This cross consists of 2 k n + 1 2kn + 1 cubes; a central cube together with 2 n 2n arms of length k k . A family of translates of a cross whose union is n n -dimensional Euclidean space and whose interiors are disjoint is a tiling. Denote the set of translation vectors by L {\mathbf {L}} . If the vector set L {\mathbf {L}} is a vector lattice, then we say that the tiling is a lattice tiling. If every vector of L {\mathbf {L}} has rational coordinates, then we say that the tiling is a rational tiling, and, similarly, if every vector of L {\mathbf {L}} has integer coordinates, then we say that the tiling is an integer tiling. Is there a noninteger tiling by cross...
In this paper, we single out the following particular case of the clique search problem. The vert... more In this paper, we single out the following particular case of the clique search problem. The vertices of the given graph are legally colored with k colors and we are looking for a clique with k nodes in the graph. In other words, we want to decide if a given k-partite graph contains a clique with k nodes. The maximum clique problem asks for finding a maximum clique in a given finite simple graph. The problem of deciding if the given graph contains a clique with k vertices is called the k-clique problem. The first problem is NP-hard and the second one is NP-complete. The special clique search problem, we propose, is still an NP-complete problem. We will show that the k-clique problem in the special case of k-partite graphs is more tractable than in the general case. In order to illustrate the possible practical utility of this restricted type clique search problem we will show that the job shop scheduling problem can be reduced to such a clique search problem in a suitable constructe...
In 1961 J. Stein proposed an algorithm to compute the greatest common divisor of two integers. In... more In 1961 J. Stein proposed an algorithm to compute the greatest common divisor of two integers. In this paper we point out that similar algorithms exist in the ring of integers of various quadratic number fields and also in the non-commutative ring of the Hurwitz quaternions. The implementations of the algorithms are straightforward. However the procedures vary from case to case. AMS Mathematics Subject Classification (2010): Primary 11A51, Secondary 11Y16 Key words and phrases: gcd in quadratic number fields, Hurwitz quater-nion, binary gcd algorithm and its extensions.
G. Hajós proved that if a finite abelian group is a direct product of its cyclic subsets, then at... more G. Hajós proved that if a finite abelian group is a direct product of its cyclic subsets, then at least one of the factors must be a subgroup. We give a new elementary proof of this theorem based on the special case for p-groups.
The paper will present a method to establish an upper bound on the clique number of a given finit... more The paper will present a method to establish an upper bound on the clique number of a given finite simple graph. In order to evaluate the performance of the proposed algorithm in practice we carry out a large scale numerical experiment on carefully selected benchmark instances. Povzetek: Razvita in opisana je nova metoda za določanje zgornje meje števila klik v grafu.
Three infinite families of finite abeliab groups will be described such that each members of thes... more Three infinite families of finite abeliab groups will be described such that each members of these families has the Redei k-property for many non-trivial values of k.
A spectral set in R^n is a set X of finite Lebesgue measure such that L^2(X) has an orthogonal ba... more A spectral set in R^n is a set X of finite Lebesgue measure such that L^2(X) has an orthogonal basis of exponentials. It is conjectured that every spectral set tiles R^n by translations. A set of translations T has a universal spectrum if every set that that tiles by translations by T has this spectrum. A recent result proved that many periodic tiling sets have universal spectra, using results from factorizations of abelian groups, for groups for which a strong form of a conjecture of Tijdeman is valid. This paper shows Tijdeman's conjecture does not hold for the cyclic group of order 900. It formulates a new sufficient conjecture for a periodic tiling set to have a universal spectrum, and uses it to show that the tiling sets for the counterexample above do have universal spectra.
By a theorem of L. Rédei if a finite abelian group is a direct product of its subsets such that e... more By a theorem of L. Rédei if a finite abelian group is a direct product of its subsets such that each subset has a prime number of elements and contains the identity element of the group, then at least one of the factors must be a subgroup. The content of this paper is that this result holds for certain infinite abelian groups, too. Namely, for groups that are direct products of finitely many Prüferian groups and finite cyclic groups of prime power order, belonging to pairwise distinct primes.
The paper deals with the following problem: If a finite abelian 2-group is a direct product of it... more The paper deals with the following problem: If a finite abelian 2-group is a direct product of its subsets of cardinality 4, does it follow that at least one of the factors is periodic? Two results are presented. In the first one, the structures of the group and the subsets are restricted but the size of the the group is not. In the second one, the group and the factors are general but the order of the group is 26.
If |A1|= q1, . . . , |An|= qn, then (q1, . . . ,qn) is called the type of the factorization A1 · ... more If |A1|= q1, . . . , |An|= qn, then (q1, . . . ,qn) is called the type of the factorization A1 · · ·An. By the fundamental theorem of finite abelian groups each finite abelian group is a direct product of cyclic groups. This decomposition into cyclic groups is not necessarily unique. However, if G is the direct product of cyclic groups of orders t1, . . . , tr respectively, then we say that G is of type (t1, . . . , tr). We use
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