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ABSTRACT: Given a fixed positive integer k ≥ 2, let G be a simple graph of order n ≥ 6k. It is proved that if the minimum degree of G is at least n/2 + 1, then for every pair of vertices x and y, there exists a Hamiltonian cycle such that the distance between x and y along that cycle is precisely k.Graphs and Combinatorics 07/2014; 30(4). DOI:10.1007/s0037301313259 · 0.33 Impact Factor 
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). 
Article: Language Reference
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ABSTRACT: Nonpath spectrum sets. (English summary)  [Show abstract] [Hide abstract]
ABSTRACT: We prove that any cycle01/2008;  SIAM Journal on Discrete Mathematics 01/2007; 21(1). DOI:10.1137/S0895480102414107 · 0.58 Impact Factor
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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. DOI:10.1002/jgt.3190120212 · 0.67 Impact Factor  Ars Combinatoria Waterloo then Winnipeg 01/2006; 79. · 0.20 Impact Factor
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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 01/2006; 51(1):22  32. DOI:10.1002/jgt.20118 · 0.67 Impact Factor  Discrete Mathematics 01/2002; 249(1). DOI:10.1016/S0012365X(01)002230 · 0.57 Impact Factor
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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. DOI:10.1016/S0012365X(00)003125 · 0.57 Impact Factor  [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...  [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(4):399410. DOI:10.1007/s003730070004 · 0.33 Impact Factor  Combinatorica 01/1999; · 0.63 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...  [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 ... 
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. DOI:10.1016/0012365X(95)00179Z · 0.57 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. DOI:10.1016/0012365X(94)902585 · 0.57 Impact Factor  Discrete Applied Mathematics 07/1993; 44(13):191203. · 0.68 Impact Factor
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ABSTRACT: A hypcrgraph is irregular if no two of its vertices have t11e same degree. It is shown for all r 2 3 and n 2 r + 3, that there exist irregular runiform hypergraphs of order n. For r 2 6 it is proved that almost allruniform hypergraphs are irregular. A linear upper bound is given for the irregularity strength of hypergraphs of order nand fixed rank. Furthemwre the irregularity strength of complete and complete equipartite hypcrgraphs is determined.01/1992; 11.  [Show abstract] [Hide abstract]
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. DOI:10.1016/0012365X(91)90161T · 0.57 Impact Factor
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