Publications (10)4.34 Total impact
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ABSTRACT: The problem of finding a deepest point (a ball centre) of a polyhedron is studied. A finite combinatorial interior point method is presented for this problem which yields an algorithm for linear programming. We conjecture that this is a strongly polynomial algorithm. Meanwhile developing the algorithm, several auxiliary results were found; among others, Gorokh and Werner’s algorithm for linear inequalities is slightly extended. Our numerical experiments with the problem detected bugs in a linear interior point solver used in MATLAB 6 Optimization Toolbox.Optimization 01/2013; 64(2):409427. DOI:10.1080/02331934.2012.756877 · 0.77 Impact Factor 
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ABSTRACT: It is known that Moore digraphs of degree d ? 1 and diameter k ? 1 do not exist(see [20] or [5]). Furthermore, for degree 2, it is shown that for k 3 there are nodigraphs of order `close" to, i.e., one less than, Moore bound [18]. In this paper, weshall consider digraphs of diameter k, degree 3 and number of vertices one less thanMoore bound. We give a necessary condition for the existence of such digraphs and,using this condition, we deduce that such digraphs do not exist for infinitely ...  [Show abstract] [Hide abstract]
ABSTRACT: The nonexistence of digraphs with order equal to the Moore bound Mddk = 1 + d+: : : +d k for dd k > 1 has lead to the study of the problem of the existence oalmost' Moore digraphs, namely digraphs with order close to the Moore bound. In 1], it was shown that almost Moore digraphs of order Mddk ; 1, degree d, diameter k (dd k 3) contain either no cycle of length k or exactly one such cycle. In this paper we shall derive some further necessary conditions for the existence of almost Moore digraphs for degree and diameter greater than 1. As a consequence, for diameter 2 and degree d, 2 d 12, we show that there are no almost Moore digraphs of order Mdd2 ; 1 with one vertex in a C2 except the digraphs with every vertex in C2 .Ars Combinatoria Waterloo then Winnipeg 01/2000; 56. · 0.20 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We introduce the problem of finding a maximum weight matching in a graph such that the number of matched vertices lies in a prescribed interval and certain vertices will be matched. In the case of bipartite graphs, this generalizes the kcardinality assignment problem which was recently studied by Dell'Amico and Martello (Discrete Appl. Math. 76 (1997) 103–121). Similarly defined a minimum weight constrained edge covering problem is shown to be NPhard even for bipartite graphs. We present a simple polynomial transformations of such matching and simplified covering problems to classical unconstrained problems. In the case of bipartite graphs also mincost flow formulations are given.Discrete Applied Mathematics 06/1999; 92(23):229241. DOI:10.1016/S0166218X(99)000529 · 0.68 Impact Factor 
Article: A Note on Constructing Large Cayley Graphs of Given Degree and Diameter by Voltage Assignments.
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ABSTRACT: Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem,whichis to determine the largest order of a graph with given degree and diameter. Many currently known largest graphs of degree # 15 and diameter # 10 have been found by computer search among Cayley graphs of semidirect products of cyclic groups. We show that all of them can in fact be described as lifts of smaller Cayley graphs of cyclic groups, with voltages in (other) cyclic groups. # This research started when J. Plesnk and J. Siran were visiting the Department of Computer Science and Software Engineering of the University of Newcastle NSW Australia in 1995, supported by small ARC grant. 1 the electronic journal of combinatorics 5 (1998), #R9 2 This opens up a new possible direction in the search for large vertextransitive graphs of given degree and di...The electronic journal of combinatorics 01/1998; 5. · 0.57 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Finite undirected graphs (possibly with loops, parallel edges) and finite groups are considered. We denote the radius and the diameter of a graph G by ρ(G) and δ(G), respectively. From the introduction: “The aim of this paper is to draw the attention to a construction well known in topological graph theory, the covering graph construction. Roughly speaking, it enables to ((blowup)) a given base graph to a larger graph.” The construction mentioned above is analyzed in detail in Part 2 (Voltage graphs and covering spaces) of the book [J. L. Gross and T. W. Tucker, Topological graph theory (1987; Zbl 0621.05013)]. We denote by G α the “large” graph obtained from a graph G and a group Γ by use of the assignment α:E(G)→Γ. The main result of the paper is the following one. Consider a group Γ and a generating system X of Γ such that X 1 =X and X contains no idempotent element. Let H be the Cayley graph Cay(Γ,X). Whenever G is a connected graph such that any degree exceeds X (in G), then there is an α such that δ(G α )<2(ρ(G)+δ(H)).01/1998; 18.  [Show abstract] [Hide abstract]
ABSTRACT: The Moore bound for a diregular digraph of degree E5>, k and diameter k is \(\). It is known that digraphs of order \(\) do not exist for d>1 and k>1 ([24] or [6]). In this paper we study digraphs of degree E5>, k, diameter k and order \(\), denoted by (d, k)digraphs. Miller and Fris showed that (2, k)digraphs do not exist for k≥3 [22]. Subsequently, we gave a necessary condition of the existence of (3, k)digraphs, namely, (3, k)digraphs do not exist if k is odd or if k+1 does not divide \(\) [3]. The (E5>, k, 2)digraphs were considered in [4]. In this paper, we present further necessary conditions for the existence of (d, k)digraphs. In particular, for \(\), we show that a (d, k)digraph contains either no cycle of length k or exactly one cycle of length k.Graphs and Combinatorics 01/1998; 14(2):109119. DOI:10.1007/s003730050019 · 0.33 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: It is known tht Moore digraphs of degree d > 1 and diameter k > 1 do not exist (see [20] or [5]). Furthermore, for degree 2, it is shown tht for l ≥ 3 there are no digraphs of order “close” to, i.e., one less than Moore bound [18]. In this paper, we shall consider digraphs of diameter k, degree 3 and number of vertices one less than Moore bound. We give a necessary condition for the existence of such digraphs and, using this condition, we deduce that such digraphs do not exist for infinitely many values of the diameter. © 1995 John Wiley & Sons, Inc.Journal of Graph Theory 11/1995; 20(3):339  349. DOI:10.1002/jgt.3190200310 · 0.67 Impact Factor 
Article: Professor Štefan Znám (19361993)
Mathematica Slovaca 01/1994; · 0.45 Impact Factor
Publication Stats
92  Citations  
4.34  Total Impact Points  
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Institutions

1995–2013

Comenius University in Bratislava
 Department of Mathematical Analysis and Numerical Mathematics
Presburg, Bratislavský, Slovakia
