Publications

Article: Around a biclique cover conjecture
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ABSTRACT: We address an old (1977) conjecture of a subset of the authors (a variant of Ryser's conjecture): in every rcoloring of the edges of a biclique [A,B] (complete bipartite graph), the vertex set can be covered by the vertices of at most 2r2 monochromatic connected components. We reduce this conjecture to designlike conjectures, where the monochromatic components of the color classes are bicliques [X,Y] with nonempty blocks X and Y. We prove this conjecture for r<6. We show that the width (the number of bicliques) in every color class of any spanning rcoloring is at most 2^{r1} (and this is best possible).12/2012;  [Show abstract] [Hide abstract]
ABSTRACT: A graph coloring algorithm that immediately colors the vertices taken from a list without looking ahead or changing colors already assigned is called “online coloring.” The properties of online colorings are investigated in several classes of graphs. In many cases we find online colorings that use no more colors than some function of the largest clique size of the graph. We show that the first fit online coloring has an absolute performance ratio of two for the complement of chordal graphs. We prove an upper bound for the performance ratio of the first fit coloring on interval graphs. It is also shown that there are simple families resisting any online algorithm: no online algorithm can color all trees by a bounded number of colors.Journal of Graph Theory 10/2006; 12(2):217  227. · 0.63 Impact Factor  Ars Comb. 01/2006; 79.
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ABSTRACT: A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number, and Ramsey saturated otherwise. We present some conjectures and results concerning both Ramsey saturated and unsaturated graphs. In particular, we show that cycles Cn and paths Pn on n vertices are Ramsey unsaturated for all n ≥ 5. © 2005 Wiley Periodicals, Inc.Journal of Graph Theory 08/2005; 51(1):22  32. · 0.63 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Let N(n,k) be the minimum number of pairwise edge disjoint monochromatic complete graphs Kk in any 2coloring of the edges of a Kn. Upper and lower bounds on N(n,k) will be given for k⩾3. For k=3, exact values will be given for n⩽11, and these will be used to give a lower bound for N(n,3).Discrete Mathematics. 03/2001; 231(s 1–3):135–141.  [Show abstract] [Hide abstract]
ABSTRACT: Distance avoiding partitions of the Euclidean space IR n are investigated in this note. We introduce the concept of combinatorial functions of distances for point congurations, and prove a measure theoretic lemma for these functions. Using the lemma we give a new proof of a result of Larman and Rogers which had only the original combinatorial proof so far. The same lemma is used to obtain new theorems in the direction that the removal of small" sets from IR n does not decrease the unit distance chromatic number, and furthermore, that the hypothetical unit distance subgraphs of IR 2 which are not 6{colorable must have large" order. 1. Introduction. A coloring of the points of a subset A IR n using k colors is called a proper k{coloring of A if every point receives one color and no two points at unit distance apart receive the same color. The unit{distance chromatic number of A IR n is the minimum integer k such that A has a proper k{coloring. In other words, the unit...07/2000;  [Show abstract] [Hide abstract]
ABSTRACT: It will be shown that if G is a graph of order n which contains a triangle, a cycle of length n or n−1 and at least cn odd cycles of different lengths for some positive constant c, then there exists some positive constant k=k(c) such that G contains at least kn 1/6 even cycles of different lengths. Other results on the number of even cycle lengths which appear in graphs with many different odd length cycles will be given.Graphs and Combinatorics 01/2000; 16:399410. · 0.35 Impact Factor  Combinatorica 01/1999; · 0.56 Impact Factor

Article: The Visibility Number Of A Graph
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ABSTRACT: . We introduce the visibility number b(G) of a graph G, which is the minimum t such that G can be represented by assigning each vertex a union of at most t horizontal segments in the plane so that vertices u; v are adjacent if and only if some point assigned to u sees some point assigned to v via a vertical segment unobstructed by other assigned points. We prove the following: 1) every planar graph has visibility number at most 2, which is sharp. 2) r b(K m;n ) r + 1, where r = d(mn + 4)=(2m + 2n)e. 3) dn=6e b(K n ) dn=6e + 1. 4) When G has n vertices, b(G) dn=6e + 2. 1. INTRODUCTION Researchers in computational geometry have studied the use of graphs to model visibility relations in the plane. For example, in a polygon in the plane we say that two vertices "see" each other if the segment joining them lies inside the polygon. Letting vertices that see each other be adjacent defines the visibility graph of the polygon. Similarly, we can define a visibility graph on a set of line s...08/1998;  [Show abstract] [Hide abstract]
ABSTRACT: In the area of online algorithms the existence of competitive algorithms is one of the most frequently studied questions. However, competitive online graph coloring algorithms exist only for very restricted families of graphs. We introduce a new concept, called online competitive algorithms. Our main problem is whether online competitive coloring algorithms exist for all classes of graphs. This concept can be useful if somebody must design an online coloring algorithm and the input graph is only known to be in a specified class of graphs. In this case the designer want to get the best algorithm but this is usually hard. An online competitive algorithm offers less: it comes together with a function f such that for every graph in the class the number of colors it uses can be bounded by f(Ø (G)) where Ø (G) is the minimum number of colors can be achieved at all for that graph by any online algorithm (that algorithm may know the graph in advance). Of course, the smaller is f ...12/1997; 
Article: Graph spectra
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ABSTRACT: The kspectrum sk(G) of a graph G is the set of all positive integers that occur as the size of an induced kvertex subgraph of G. In this paper we determine the minimum order and size of a graph G with sk (G) = {0, 1, …,(2k)} and consider the more general question of describing those sets S ⊆ {0,1, … ,(2k)} such that S = sk(G) for some graph G.Discrete Mathematics 04/1996; 150(s 1–3):103–113. · 0.58 Impact Factor 
Article: On the rotation distance of graphs
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ABSTRACT: Let (x, y) be an edge of a graph G. Then the rotation of (x, y) about x is the operation of removing (x, y) from G and inserting (x, y′) as an edge, where y′ is a vertex of G. The rotation distance between graphs G and H is the minimum number of rotations necessary to transform G into H. Lower and upper bounds are given on the rotation distance of two graphs in terms of their greatest common subgraphs and their partial rotation link of largest cardinality. We also propose some extermal problems for the rotation distance of trees.Discrete Mathematics. 03/1994; 126(s 1–3):121–135.  Discrete Applied Mathematics 07/1993; 44(13):191203. · 0.72 Impact Factor
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ABSTRACT: It is proved that if t is a fixed positive integer and n is sufficiently large, then each graph of order n with minimum degree n − t has an assignment of weights 1, 2 or 3 to the edges in such a way that weighted degrees of the vertices become distinct.Discrete Mathematics. 08/1991; 91(1):45–59.  Journal of Graph Theory  JGT. 01/1991;
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ABSTRACT: A network is a simple graph to which each edge has been assigned a positive integer weight. A network is irregular if the sum of the edges incident to each vertex is distinct. In this paper we study this concept for regular or nearly regular graphs and derive a relationship to integer matrices with distinct row and column sums.In particular, we consider the parameter, s(G), the irregularity strength of a graph G, which is the smallest maximum weight over all irregular networks with underlying graph G. It is known that if G is an rregular graph of order n, then . We exhibit infinitely many rregular graphs with , and it is proved that , for all rregular graphs on n vertices if r is even.We also study totally irregular matrices, that is positive integer matrices with distinct row and column sums having the smallest possible maximal entry. As a corollary, we can determine the strength of complete bipartite graphs Kp,q except in the case when p=q is odd.Discrete Mathematics. 08/1989; 76(3):223–240. 
Article: Ramsey numbers for local colorings
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ABSTRACT: The concept of a localkcoloring of a graphG is introduced and the corresponding localkRamsey numberr loc k (G) is considered. A localkcoloring ofG is a coloring of its edges in such a way that the edges incident to any vertex ofG are colored with at mostk colors. The numberr loc k (G) is the minimumm for whichK m contains a monochromatic copy ofG for every localkcoloring ofK m . The numberr loc k (G) is a natural generalization of the usual Ramsey numberr k (G) defined for usualkcolorings. The results reflect the relationship betweenr k (G) andr loc k (G) for certain classes of graphs.Graphs and Combinatorics 11/1987; 3(1):267277. · 0.35 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: A local kcoloring of a graph is a coloring of its edges in such a way that each vertex is incident to edges of at most k different colors. We investigate the similarities and differences between usual and local kcolorings, and the results presented in the paper give a general insight to the nature of local coloring. We are mainly concerned with local variants of Ramseytype problems, in particular, with Ramsey's theorem for hypergraphs, the existence of minimal Ramsey graphs and further questions from noncomplete Ramsey Theory.Journal of Combinatorial Theory, Series B. 01/1987;  [Show abstract] [Hide abstract]
ABSTRACT: We study the functionb(n, d), the maximal number of atoms defined byn ddimensional boxes, i.e. parallelopipeds in theddimensional Euclidean space with sides parallel to the coordinate axes. We characterize extremal interval families definingb(n, 1)=2n1 atoms and we show thatb(n, 2)=2n 26n+7. We prove that for everyd, b* (d) = limn ® ¥ b(n,d)/ndb^* (d) = \mathop {\lim }\limits_{n \to \infty } b(n,d)/n^d exists and 1 \leqq (d/2)dÖ{b* (d)} \leqq e1 \leqq (d/2)\sqrt[d]{{b^* (d)}} \leqq e . Moreover, we obtainb*(3)=8/9.Combinatorica 08/1985; 5(3):193204. · 0.56 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The following generalizations and relatives of interval families are studied in the paper: arcs of a circle, multiple intervals, chords of a circle, ddimensional boxes and multiple boxes. For these families we survey results and problems concerning the dependence of the transversal number on the packing number and the dependence of the coloring number on the clique number.The intersection graphs of the underlying set systems are circular arc graphs, multiple interval graphs, circle graphs (called also overlap graphs), box graphs and multiple box graphs. Thus most of our problems and results concern the relation between the cliquecover number and stability number (ϑ and α), or between the chromatic number and clique number (χ and ω) of these graphs.Discrete Mathematics. 01/1985;
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