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ABSTRACT Summary: "We consider the coloring of edges in a graph in which there are vertices of three types. In a feasible edge coloring, each vertex of the first type is incident with at least two edges of the same color, and... more
ABSTRACT Summary: "We consider the coloring of edges in a graph in which there are vertices of three types. In a feasible edge coloring, each vertex of the first type is incident with at least two edges of the same color, and each vertex of the second type with at least two edges of different colors, while no constraints are required for the vertices of the third type. We present a characterization of colorable graphs and a linear-time algorithm to decide whether a given graph with prescribed vertex types admits a feasible edge coloring.''
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ABSTRACT
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We investigate the largest number of colours, called upper chromatic number and denoted X(H), that can be assigned to the vertices (points) of a Steiner triple system H in such a way that every block H ∈ H contains at least two vertices... more
We investigate the largest number of colours, called upper chromatic number and denoted X(H), that can be assigned to the vertices (points) of a Steiner triple system H in such a way that every block H ∈ H contains at least two vertices of the same colour. The exact value of X is determined for some classes of triple systems,
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The paper surveys problems, results and methods concerning the coloring of Steiner triple and quadruple systems viewed as mixed hypergraphs. In this setting, two types of conditions are considered: each block of the Steiner system in... more
The paper surveys problems, results and methods concerning the coloring of Steiner triple and quadruple systems viewed as mixed hypergraphs. In this setting, two types of conditions are considered: each block of the Steiner system in question has to contain (i) a monochromatic pair of vertices, or, more, restrictively, (ii) a triple of vertices that meets precisely two color classes.
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ELSEVIER Discrete Mathematics 174 (1997) 247259 DISCRETE MATHEMATICS Upper chromatic number of Steiner triple and quadruple systems Lorenzo Milazzoa'*, Zsolt Tuzab a Department of Mathematics, University of Catania, viale A. Doria,... more
ELSEVIER Discrete Mathematics 174 (1997) 247259 DISCRETE MATHEMATICS Upper chromatic number of Steiner triple and quadruple systems Lorenzo Milazzoa'*, Zsolt Tuzab a Department of Mathematics, University of Catania, viale A. Doria, 6, 95125 Catania, Italy b Computer ...
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We study the problem of scheduling groups of tasks with precedence constraints on three dedicated processors. Each task requires a specified set of processors. Up to three precedence constraints are considered among groups of tasks... more
We study the problem of scheduling groups of tasks with precedence constraints on three dedicated processors. Each task requires a specified set of processors. Up to three precedence constraints are considered among groups of tasks requiring the same set of processors. The objective of the problem is to find a nonpreemptive schedule which minimizes the maximum completion time (makespan). This
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ABSTRACT We survey results and open problems on ‘mixed hypergraphs’ that are hypergraphs with two types of edges. In a proper vertex coloring the edges of the first type must not be monochromatic, while the edges of the second type must... more
ABSTRACT We survey results and open problems on ‘mixed hypergraphs’ that are hypergraphs with two types of edges. In a proper vertex coloring the edges of the first type must not be monochromatic, while the edges of the second type must not be completely multicolored. Though the first condition just means ‘classical’ hypergraph coloring, its combination with the second one causes rather unusual behavior. For instance, hypergraphs occur that are uncolorable, or that admit colorings with certain numbers k′ and k″ of colors but no colorings with exactly k colors for any k′ < k < k″.
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We survey recent developments and open problems on graph coloringswhere the color of each vertex has to be chosen from a restrictedset of admissible colors.1 PreliminariesGraph colorings belong to classical graph theoretical problems that... more
We survey recent developments and open problems on graph coloringswhere the color of each vertex has to be chosen from a restrictedset of admissible colors.1 PreliminariesGraph colorings belong to classical graph theoretical problems that are importantboth for their practical applications and richness of theoretical results.E.g., the Four Color Conjecture has stimulated research in discretemathematics for more than hundred years, and
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We treat a variation of domination which involves a partition V = (V1,V2,...,Vk) of V (G) and domination of each partition class Vi over distance d where all vertices and edges of G may be used in the domination process. Strict upper... more
We treat a variation of domination which involves a partition V = (V1,V2,...,Vk) of V (G) and domination of each partition class Vi over distance d where all vertices and edges of G may be used in the domination process. Strict upper bounds and extremal graphs are presented; the results are collected in three handy tables. Further, we compare a
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We prove that for every complete multipartite graph $F$ there exist very dense graphs $G_n$ on $n$ vertices, namely with as many as ${n\choose 2}-cn$ edges for all $n$, for some constant $c=c(F)$, such that $G_n$ can be decomposed into... more
We prove that for every complete multipartite graph $F$ there exist very dense graphs $G_n$ on $n$ vertices, namely with as many as ${n\choose 2}-cn$ edges for all $n$, for some constant $c=c(F)$, such that $G_n$ can be decomposed into edge-disjoint induced subgraphs isomorphic to~$F$. This result identifies and structurally explains a gap between the growth rates $O(n)$ and $\Omega(n^{3/2})$ on the minimum number of non-edges in graphs admitting an induced $F$-decomposition.
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ABSTRACT We introduce and study a Maker-Breaker type game in which the issue is to create or avoid two disjoint dominating sets in graphs without isolated vertices. We prove that the maker has a winning strategy on all connected graphs if... more
ABSTRACT We introduce and study a Maker-Breaker type game in which the issue is to create or avoid two disjoint dominating sets in graphs without isolated vertices. We prove that the maker has a winning strategy on all connected graphs if the game is started by the breaker. This implies the same in the $(2:1)$ biased game also in the maker-start game. It remains open to characterize the maker-win graphs in the maker-start non-biased game, and to analyze the $(a:b)$ biased game for $(a:b)\neq (2:1)$. For a more restricted variant of the non-biased game we prove that the maker can win on every graph without isolated vertices.
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Invited talk at the International Conference Combinatorics 2012: 1912-2012: �One hundred years to Chromatic Polynomial. Perugia, Italy, 2012.
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We prove that if one or more players in a locally finite positional game have winning strategies, then they can find it by themselves, not losing more than a bounded number of plays and not using more than a linear-size memory,... more
We prove that if one or more players in a locally finite positional game have winning strategies, then they can find it by themselves, not losing more than a bounded number of plays and not using more than a linear-size memory, independently of the strategies applied by the other players. We design two algorithms for learning how to win. One
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ABSTRACT The choice ratio of a graph G = (V; E) is the minimum quotient a=b with the following property. For every assignment of a-element sets L(v) to the vertices v 2 V , there can be chosen b-element subsets C(v) ae L(v) for all v in... more
ABSTRACT The choice ratio of a graph G = (V; E) is the minimum quotient a=b with the following property. For every assignment of a-element sets L(v) to the vertices v 2 V , there can be chosen b-element subsets C(v) ae L(v) for all v in such a way that C(v) " C(v 0 ) = ; holds for every adjacent vertex pair v; v 0 . Alon, Tuza and Voigt [Discr. Math. 165/166, 1997, 31--38.] proved that the choice ratio is equal to the fractional chromatic number, for every graph G. In this paper we prove that the analogous statement holds for the fractional P-chromatic number and the P-choice-ratio of G, where P is any hereditary graph property. A generalization of these results for hypergraphs is presented, too. Keywords: graph property, vertex partition, fractional P-coloring, P-choice ratio, hypergraph coloring. 1991 Mathematics Subject Classification: 05C15, O5C75 1 1 Introduction In this paper we investigate the combination of two subjects of graph theory, both of which have attrected mu...
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... Satisfactory Partitions in a Graph Cristina Bazgan1, Zsolt Tuza2, and Daniel Vanderpooten1 ...Chvátal introduced in [2] the decomposition problem of bicoloring the vertices of a graph in such a way that each vertex has at most one... more
... Satisfactory Partitions in a Graph Cristina Bazgan1, Zsolt Tuza2, and Daniel Vanderpooten1 ...Chvátal introduced in [2] the decomposition problem of bicoloring the vertices of a graph in such a way that each vertex has at most one neighbor with a different color. ...
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The Satisfactory Partition problem asks for deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part. This problem was... more
The Satisfactory Partition problem asks for deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part. This problem was introduced by Gerber and Kobler [M. Gerber, D. Kobler, Algorithmic approach to the satisfactory graph partitioning problem,
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In this paper we consider the problem of semi-online scheduling on two uniform processors, in the case where the total sum of the tasks is known in advance. Tasks arrive one at a time and have to be assigned to one of the two processors... more
In this paper we consider the problem of semi-online scheduling on two uniform processors, in the case where the total sum of the tasks is known in advance. Tasks arrive one at a time and have to be assigned to one of the two processors before the next one arrives. The assignment cannot be changed later. The objective is the minimization of the makespan. Assume that the speed of the fast processor is s, while the speed of the slow one is normalized to 1. As a function of s, we derive general lower bounds on the competitive ratio achievable with respect to offline optimum, and design on-line algorithms with guaranteed upper bound on their competitive ratio. The algorithms presented for s≥3 are optimal, as well as for s=1 and for 1+174≤s≤1+32.