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Circular colorings of edgeweighted graphs
 J. Graph Theory
, 2003
"... The notion of (circular) colorings of edgeweighted graphs is introduced. This notion generalizes the notion of (circular) colorings of graphs, the channel assignment problem, and several other optimization problems. For instance, its restriction to colorings of weighted complete graphs corresponds ..."
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The notion of (circular) colorings of edgeweighted graphs is introduced. This notion generalizes the notion of (circular) colorings of graphs, the channel assignment problem, and several other optimization problems. For instance, its restriction to colorings of weighted complete graphs corresponds to the traveling salesman problem (metric case). It also gives rise to a new definition of the chromatic number of directed graphs. Several basic results about the circular chromatic number of edgeweighted graphs are derived. 1
Hajós theorem for list coloring of hypergraphs
, 2001
"... A well known theorem Hajós claims that every graph with chromatic number greater than k can be constructed from disjoint copies of the complete graph K k+1 by repeated application of three simple operations. This classical result has been extended in 1978 to coloring the hypergraphs by C. Benzaken a ..."
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Cited by 1 (0 self)
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A well known theorem Hajós claims that every graph with chromatic number greater than k can be constructed from disjoint copies of the complete graph K k+1 by repeated application of three simple operations. This classical result has been extended in 1978 to coloring the hypergraphs by C. Benzaken and in 1996 to list coloring graphs by S. Gravier. In this note, we capture both variations to extend Hajós theorem to list coloring hypergraphs.
A Hajóslike Theorem for Weighted Coloring
 JOURNAL OF THE BRAZILIAN COMPUTER SOCIETY
, 2013
"... The Hajós’ Theorem [8] shows a necessary and sufficient condition for the chromatic number of a given graph G be at least k: G must contain a kconstructible subgraph. A graph is kconstructible if it can be obtained from a complete graph of order k by successively applying a set of welldefined op ..."
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The Hajós’ Theorem [8] shows a necessary and sufficient condition for the chromatic number of a given graph G be at least k: G must contain a kconstructible subgraph. A graph is kconstructible if it can be obtained from a complete graph of order k by successively applying a set of welldefined operations. Given a vertexweighted graph G and a (proper) rcoloring c = {C1,..., Cr} of G, the weight of a color class Ci is the maximum weight of a vertex colored i and the weight of c is the sum of the weights of its color classes. The objective of the Weighted Coloring Problem [7] is, given a vertexweighted graph G, to determine the minimum weight of a proper coloring of G, that is its weighted chromatic number. In this article, we prove that the Weighted Coloring Problem admits a version of the Hajós ’ Theorem and so we show a necessary and sufficient condition for the weighted chromatic number of a vertexweighted graph G be at least k, for any positive real k.
Coloring Weighted SeriesParallel Graphs
, 2003
"... Let G be a seriesparallel graph with integer edge weights. A pcoloring of G is a mapping of vertices of G into Zp (ring of integers modulo p) so that the distance between colors of adjacent vertices u and v is at least the weight of the edge uv. We describe a quadratic time pcoloring algorithm wh ..."
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Let G be a seriesparallel graph with integer edge weights. A pcoloring of G is a mapping of vertices of G into Zp (ring of integers modulo p) so that the distance between colors of adjacent vertices u and v is at least the weight of the edge uv. We describe a quadratic time pcoloring algorithm where p is either twice the maximum edge weight or the largest possible sum of three weights of edges lying on a common cycle. Povzetek: Opisano je barvanje grafov. 1
and
"... A wellknown theorem of Hajós claims that every graph with chromathic number greater than k can be constructed from disjoint copies of the complete graph Kk+1 by repeated application of three simple operations. This classical result has been extended in 1978 to colorings of hypergraphs by C. Benza ..."
Abstract
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A wellknown theorem of Hajós claims that every graph with chromathic number greater than k can be constructed from disjoint copies of the complete graph Kk+1 by repeated application of three simple operations. This classical result has been extended in 1978 to colorings of hypergraphs by C. Benzaken and in 1996 to listcolorings of graphs by S. Gravier. In this note, we capture both variations to extend Hajós’ theorem to listcolorings of hypergraphs.