Results 1  10
of
32
ON THE COMBINATORIAL PROBLEMS WHICH I WOULD MOST LIKE TO SEE SOLVED
, 1979
"... I was asked to write a paper about the major unsolved problems in combinatorial mathematics. After some thought it seemed better to modify the title to a less pretentious one. Combinatorial mathematics has grown enormously and a genuine survey would have to include not only topics where I have no re ..."
Abstract

Cited by 23 (0 self)
 Add to MetaCart
I was asked to write a paper about the major unsolved problems in combinatorial mathematics. After some thought it seemed better to modify the title to a less pretentious one. Combinatorial mathematics has grown enormously and a genuine survey would have to include not only topics where I have no real competence but also topics about which I never seriously thought, e.g. algorithmic combinatorics, coding theory and matroid theory. There is no doubt that the proof of the conjecture that several simply stated problems have no good algorithm is fundamental and may have important consequences for many other branches of mathematics, but unfortunately I have no real feeling for these questions and I feel I should leave the subject to those who are more competent. I just heard that Khachiyan [59], has a polynomial algorithm for linear programming. (See also [50].) This is considered a sensational result and during my last stay in the U.S. many of my friends were greatly impressed by it.
Linearity and solvability in multicast networks
 IEEE TRANS. INF. THEORY
, 2004
"... It is known that for every solvable multicast network, there exists a large enough finitefield alphabet such that a scalar linear solution exists. We prove: i) every binary solvable multicast network with at most two messages has a binary scalar linear solution; ii) for more than two messages, not ..."
Abstract

Cited by 12 (3 self)
 Add to MetaCart
It is known that for every solvable multicast network, there exists a large enough finitefield alphabet such that a scalar linear solution exists. We prove: i) every binary solvable multicast network with at most two messages has a binary scalar linear solution; ii) for more than two messages, not every binary solvable multicast network has a binary scalar linear solution; iii) a multicast network that has a solution for a given alphabet might not have a solution for all larger alphabets.
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 ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
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
Mutually Orthogonal Latin Squares: A Brief Survey of Constructions
, 1999
"... In the two centuries since Euler first asked about mutually orthogonal latin squares, substantial progress has been made. The biggest breakthroughs came in 1960 with the celebrated theorems of Bose, Shrikhande, and Parker, and in 1974 in the research of Wilson. Current efforts have concentrated on r ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
In the two centuries since Euler first asked about mutually orthogonal latin squares, substantial progress has been made. The biggest breakthroughs came in 1960 with the celebrated theorems of Bose, Shrikhande, and Parker, and in 1974 in the research of Wilson. Current efforts have concentrated on refining these approaches, and finding new applications of the substantial theory opened. This paper provides a detailed list of constructions for MOLS, concentrating on the uses of pairwise balanced designs and transversal designs in recursive constructions as pioneered in the papers of Bose, Shrikhande, and Parker. In addition, several new lower bounds for MOLS are given and an uptodate table of lower bounds for MOLS is provided. 1 An Historical Introduction In 1779, Euler began a study of a simple mathematical puzzle, the 36 Officers Problem. Thirtysix officers drawn from six different ranks and six different regiments (one of each rank from each regiment) are to be arranged in a squar...
Candelabra Systems And Designs
"... . Combinatorial structures called candelabra systems can be used in recursive constructions to build Steiner 3designs. We introduce a new closure operation on natural numbers involving candelabra systems. This new closure operation makes it possible to generalize various constructions for Stein ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
. Combinatorial structures called candelabra systems can be used in recursive constructions to build Steiner 3designs. We introduce a new closure operation on natural numbers involving candelabra systems. This new closure operation makes it possible to generalize various constructions for Steiner 3designs and to create new infinite families of Steiner 2designs and 3designs. We provide an independent proof for Wilson's "product theorem" for Steiner 3designs. We also construct new group divisible designs of strength 2 and 3. AMS Classification: 05B10, 62K10 Keywords : Steiner Designs; Candelabra System; Lattice Candelabra System; Lattice Group Divisible Design; Closure Operation Date: July 8, 1999. 0 1. Introduction The problems that are of interest in design theory include determining all quadruples (t; v; k; ) for which a t(v; k; ) design exists. There are many known 2designs, but constructing tdesigns for t ? 2 is much harder, especially with = 1 (Steiner desig...
Switching of edges in strongly regular graphs. I. A family of partial difference sets on 100 vertices
 ELECTRON. J. COMBIN., 10(1):RESEARCH PAPER
, 2003
"... We present 15 new partial difference sets over 4 nonabelian groups of order 100 and 2 new strongly regular graphs with intransitive automorphism groups. The strongly regular graphs and corresponding partial difference sets have the following parameters: (100,22,0,6), (100,36,14,12), (100,45,20,2 ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
We present 15 new partial difference sets over 4 nonabelian groups of order 100 and 2 new strongly regular graphs with intransitive automorphism groups. The strongly regular graphs and corresponding partial difference sets have the following parameters: (100,22,0,6), (100,36,14,12), (100,45,20,20), (100,44,18,20). The existence of strongly regular graphs with the latter set of parameters was an open question. Our method is based on combination of Galois correspondence between permutation groups and association schemes, classical Seidel's switching of edges and essential use of computer algebra packages. As a byproduct, a few new amorphic association schemes with 3 classes on 100 points are discovered.
A Construction For Infinite Families Of Steiner 3Designs
, 1999
"... Let q be a prime power and a be a positive integer such that a 2. Assume that there is a Steiner 3(a + 1; q + 1; 1) design. For every v satisfying certain arithmetic conditions we can construct a Steiner 3(va d +1; q+1; 1) design for every d sufficiently large. In the case of block size 6, when ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Let q be a prime power and a be a positive integer such that a 2. Assume that there is a Steiner 3(a + 1; q + 1; 1) design. For every v satisfying certain arithmetic conditions we can construct a Steiner 3(va d +1; q+1; 1) design for every d sufficiently large. In the case of block size 6, when q = 5, this theorem yields new infinite families of Steiner 3designs: if v is a given positive integer satisfying the necessary arithmetic conditions, for every nonnegative integer m there exists a Steiner 3(v(4 \Delta 5 m + 1) d + 1); 6; 1) for sufficiently large d.
Covering Arrays on Graphs: Qualitative Independence Graphs and Extremal Set Partition Theory
, 2005
"... I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy. (Principal Adviser) I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy. (Principal Adviser) I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy. I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and in quality, as a dissertation for the degree of Doctor of Philosophy. Approved for the University Committee on Graduate
Multicolored parallelisms of isomorphic spanning trees
 Discrete Math. Theor. Comput. Sci
"... It is proved that a complete graph on n (> 4) vertices can be properly edgecolored with n − 1 colors in such a way that the edges can be partitioned into edge disjoint multicolored isomorphic spanning trees whenever n is a power of two or five times a power of two. A spanning tree is multicolored i ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
It is proved that a complete graph on n (> 4) vertices can be properly edgecolored with n − 1 colors in such a way that the edges can be partitioned into edge disjoint multicolored isomorphic spanning trees whenever n is a power of two or five times a power of two. A spanning tree is multicolored if all n − 1 colors occur among its edges.
SuperMagic Complete kpartite Hypergraphs
 Graphs and Comb
, 2001
"... We deal with complete kpartite hypergraphs and we show that for all k>1 and n different from 2, 6 its hyperedges can be labeled by consecutive integers 1,2,......,n^k such that the sum of labels of the hyperedges incident to (k1) particular vertices is the same for all (k1)tuples of vertices fr ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We deal with complete kpartite hypergraphs and we show that for all k>1 and n different from 2, 6 its hyperedges can be labeled by consecutive integers 1,2,......,n^k such that the sum of labels of the hyperedges incident to (k1) particular vertices is the same for all (k1)tuples of vertices from (k1) independent sets. This result was improved (for n=6) in M.Trenkler: Magic pdimensional cubes, Acta Arithmetica 96}(2001), 361364.