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Polychromatic Cliques and Related Questions
, 2003
"... Let the edges of a graph G be coloured so that no colour is used more than k times. We refer to this as a kbounded colouring. We say that a subset of the edges of G is polychromatic if each edge is of a different colour. In this paper we address the problem of finding the minimum number m such that ..."
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Let the edges of a graph G be coloured so that no colour is used more than k times. We refer to this as a kbounded colouring. We say that a subset of the edges of G is polychromatic if each edge is of a different colour. In this paper we address the problem of finding the minimum number m
1COPPE, Universidade Federal do Rio de Janeiro
"... A clique of a graph is a maximal set of vertices of size at least 2 that induces a complete graph. A kcliquecolouring of a graph is a colouring of the vertices with at most k colours such that no clique is monochromatic. Défossez proved that the 2cliquecolouring of perfect graphs is a ΣP2compl ..."
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A clique of a graph is a maximal set of vertices of size at least 2 that induces a complete graph. A kcliquecolouring of a graph is a colouring of the vertices with at most k colours such that no clique is monochromatic. Défossez proved that the 2cliquecolouring of perfect graphs is a ΣP2
Turán’s theorem in the hypercube
 SIAM Journal on Discrete Mathematics
"... We are motivated by the analogue of Turán’s theorem in the hypercube Qn: how many edges can a Qdfree subgraph of Qn have? We study this question through its Ramseytype variant and obtain asymptotic results. We show that for every odd d it is possible to color the edges of Qn with (d+1)2 4 colors, ..."
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Cited by 12 (1 self)
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, such that each subcube Qd is polychromatic, that is, contains an edge of each color. The number of colors is tight up to a constant factor, as it turns out that a similar coloring with � � d+1 2 + 1 colors is not possible. The corresponding question for vertices is also considered. It is not possible to color
Mixed Hypergraphs and Other Coloring Problems
, 2003
"... A mixed hypergraph is a triple (V; C; D) where V is the vertex set and C and D are sets of subsets of V called Cedges and Dedges, respectively. A proper coloring of a mixed hypergraph (V; C; D) is a coloring of its vertices such that no Cedge is polychromatic and no Dedge is monochromatic. W ..."
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Cited by 3 (0 self)
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A mixed hypergraph is a triple (V; C; D) where V is the vertex set and C and D are sets of subsets of V called Cedges and Dedges, respectively. A proper coloring of a mixed hypergraph (V; C; D) is a coloring of its vertices such that no Cedge is polychromatic and no Dedge is monochromatic
On the Upper Chromatic Number of a Hypergraph
, 1995
"... We introduce the notion of a coedge of a hypergraph, which is a subset of vertices to be colored so that at least two vertices are of the same color. Hypergraphs with both edges and coedges are called mixed hypergraphs. The maximal number of colors for which there exists a mixed hypergraph colorin ..."
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Cited by 27 (8 self)
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We introduce the notion of a coedge of a hypergraph, which is a subset of vertices to be colored so that at least two vertices are of the same color. Hypergraphs with both edges and coedges are called mixed hypergraphs. The maximal number of colors for which there exists a mixed hypergraph coloring using all the colors is called the upper chromatic number of a hypergraph H and is denoted by (H). An algorithm for computing the number of colorings of a mixed hypergraph is proposed. The properties of the upper chromatic number and the colorings of some classes of hypergraphs are discussed. A greedy polynomial time algorithm for finding a lower bound for (H) of a hypergraph H containing only coedges is presented.
Open Access Of woods and webs: Possible alternatives to the tree of life for studying genomic fluidity in E. coli
"... Background: We introduce several forestbased and networkbased methods for exploring microbial evolution, and apply them to the study of thousands of genes from 30 strains of E. coli. This case study illustrates how additional analyses could offer fast heuristic alternatives to standard tree of lif ..."
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of life (TOL) approaches. Results: We use gene networks to identify genes with atypical modes of evolution, and genome networks to characterize the evolution of genetic partnerships between E. coli and mobile genetic elements. We develop a novel polychromatic quartet method to capture patterns
Backbone Colorings for Networks
, 2003
"... We introduce and study backbone colorings, a variation on classical vertex colorings: Given a graph G = (V, E) and a spanning subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex coloring V > {1, 2, ...} of G in which the colors assigned to adjacent vertices in ..."
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Cited by 7 (3 self)
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We introduce and study backbone colorings, a variation on classical vertex colorings: Given a graph G = (V, E) and a spanning subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex coloring V > {1, 2, ...} of G in which the colors assigned to adjacent vertices in H differ by at least two. We study the cases where the backbone is either a spanning tree or a spanning path. We show that for tree backbones of G...
Backbone Colorings for Graphs: Tree and Path Backbones
 JOURNAL OF GRAPH THEORY
, 2007
"... We introduce and study backbone colorings, a variation on classical vertex colorings: Given a grap hG = (V,E) and a spanning subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex coloring V → {1,2,...} of G in which the colors assigned to adjacent vertices in H dif ..."
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Cited by 4 (0 self)
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We introduce and study backbone colorings, a variation on classical vertex colorings: Given a grap hG = (V,E) and a spanning subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex coloring V → {1,2,...} of G in which the colors assigned to adjacent vertices in H differ by at least two. We study the cases where the backbone is either a spanning tree or a spanning path. We show that for tree backbones of G the number of colors needed for a backbone coloring of G can roughly differ by a multiplicative factor of at most 2 from the chromatic number χ(G); for path backbones this factor is roughly 32. We show that the computational complexity of the problem “Given a graph G, a spanning tree T of G, and an integer , is there a backbone coloring for G and T with at most colors? ” jumps from polynomial to NPcomplete between = 4 (easy for all spanning trees) and = 5 (difficult even for spanning paths). We finish
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