Igor Fabrici, Frank Göring (2016)
Discussiones Mathematicae Graph Theory
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A unique-maximum k-coloring with respect to faces of a plane graph G is a coloring with colors 1, . . . , k so that, for each face of G, the maximum color occurs exactly once on the vertices of α. We prove that any plane graph is unique-maximum 3-colorable and has a proper unique-maximum coloring with 6 colors.
Javier Barrallo, Santiago Sanchez (2001)
Visual Mathematics
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András Hajnal (2008)
Fundamenta Mathematicae
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Given a function f, a subset of its domain is a rainbow subset for f if f is one-to-one on it. We start with an old Erdős problem: Assume f is a coloring of the pairs of ω₁ with three colors such that every subset A of ω₁ of size ω₁ contains a pair of each color. Does there exist a rainbow triangle? We investigate rainbow problems and results of this style for colorings of pairs establishing negative "square bracket" relations.
Dennis Geller, Hudson Kronk (1974)
Fundamenta Mathematicae
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Richard H. Schelp (2002)
Discussiones Mathematicae Graph Theory
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The focus of this article is on three of the author's open conjectures. The article itself surveys results relating to the conjectures and shows where the conjectures are known to hold.
Piotr Borowiecki (2006)
Discussiones Mathematicae Graph Theory
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For a given induced hereditary property 𝓟, a 𝓟-coloring of a graph G is an assignment of one color to each vertex such that the subgraphs induced by each of the color classes have property 𝓟. We consider the effectiveness of on-line 𝓟-coloring algorithms and give the generalizations and extensions of selected results known for on-line proper coloring algorithms. We prove a linear lower bound for the performance guarantee function of any stingy on-line 𝓟-coloring algorithm. In the...
J. Czap, S. Jendrol’, J. Valiska (2017)
Discussiones Mathematicae Graph Theory
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Given three planar graphs F,H, and G, an (F,H)-WORM coloring of G is a vertex coloring such that no subgraph isomorphic to F is rainbow and no subgraph isomorphic to H is monochromatic. If G has at least one (F,H)-WORM coloring, then W−F,H(G) denotes the minimum number of colors in an (F,H)-WORM coloring of G. We show that (a) W−F,H(G) ≤ 2 if |V (F)| ≥ 3 and H contains a cycle, (b) W−F,H(G) ≤ 3 if |V (F)| ≥ 4 and H is a forest with Δ (H) ≥ 3, (c) W−F,H(G) ≤ 4 if |V (F)| ≥ 5 and H is...
Axenovich, Maria, Manske, Jacob (2008)
Integers
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Ward, C., Szabó, S. (1994)
Acta Mathematica Universitatis Comenianae. New Series
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Exoo, Geoffrey (1998)
The Electronic Journal of Combinatorics [electronic only]
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Philippe Meurdesoif, Benoît Rottembourg (2001)
RAIRO - Operations Research - Recherche Opérationnelle
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In this paper we will describe a new class of coloring problems, arising from military frequency assignment, where we want to minimize the number of distinct -uples of colors used to color a given set of -complete-subgraphs of a graph. We will propose two relaxations based on Semi-Definite Programming models for graph and hypergraph coloring, to approximate those (generally) NP-hard problems, as well as a generalization of the works of Karger et al. for hypergraph coloring, to find...
Sabo, Dusty, Schaal, Daniel, Tokaz, Jacent (2007)
Integers
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P. Francis, S. Francis Raj (2016)
Discussiones Mathematicae Graph Theory
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A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. In this paper, we obtain bounds for the b- chromatic number of induced subgraphs in terms of the b-chromatic number of the original graph. This turns out to be...
Jungić, Veselin, Nešetřil, Jaroslav, Radoičić, Radoš (2005)
Integers
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