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Approximating the chromatic index of multigraphs
"... It is well known that if G is a multigraph then χ ′(G) ≥ χ ′∗(G): = max{∆(G), Γ(G)}, where χ ′(G) is the chromatic index of G, χ ′∗(G) is the fractional chromatic index of G, ∆(G) is the maximum degree of G, and Γ(G) = max{2E(G[U])/(U  − 1) : U ⊆ V (G), U  ≥ 3, U  is odd}. The conjecture ..."
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this conjecture for multigraphs G with χ ′(G)> ⌊(11∆(G) + 8)/10⌋; and Scheide recently improved this bound to χ ′(G)> ⌊(15∆(G) + 12)/14⌋. We prove this conjecture for multigraphs G with χ ′(G)> ⌊∆(G) + √ ∆(G)/2⌋, improving the above mentioned results. Our proof yields an algorithm for edgecoloring any
EdgeColoring SeriesParallel Multigraphs
, 2000
"... We give a simpler proof of Seymour's Theorem on edgecoloring seriesparallel multigraphs and derive a lineartime algorithm to check whether a given seriesparallel multigraph can be colored with a given number of colors. ..."
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We give a simpler proof of Seymour's Theorem on edgecoloring seriesparallel multigraphs and derive a lineartime algorithm to check whether a given seriesparallel multigraph can be colored with a given number of colors.
Satellite Communications and Multigraph EdgeColoring
"... An overview of satellite communication scheduling and its relation to edgecoloring of multigraphs is given. Then a theorem about a restricted class of multigraphs is proved to obtain conditions for scheduling in a satellite communications network of practical interest Some limitations of the multig ..."
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An overview of satellite communication scheduling and its relation to edgecoloring of multigraphs is given. Then a theorem about a restricted class of multigraphs is proved to obtain conditions for scheduling in a satellite communications network of practical interest Some limitations
Approximating Maximum Edge Coloring in Multigraphs
 In APPROX, volume 2462 of LNCS
, 2002
"... We study the complexity of the following problem that we call Max edge tcoloring: given a multigraph G and a parameter t, color as many edges as possible using t colors, such that no two adjacent edges are colored with the same color. (Equivalently, find the largest edge induced subgraph of G that ..."
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that has chromatic index at most t). We show that for every fixed t 2 there is some > 0 such that it is NPhard to approximate Max edge tcoloring within a ratio better than 1 . We design approximation algorithms for the problem with constant factor approximation ratios. An interesting feature of our
EdgeColoring Bipartite Multigraphs in O(E log D) Time
, 1999
"... Let V , E, and D denote the cardinality of the vertex set, the cardinality of the edge set, and the maximum degree of a bipartite multigraph G. We show that a minimal edgecoloring of G can be computed in O(E log D) time. 1 Introduction The edgecoloring problem is to nd a minimal edgecoloring ..."
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of an edgecoloring c of a multigraph G refers to the subset of edges having a certain color in c. An edgecoloring of a multigraph G is said to be minimal if there is no edgecoloring of G using fewer colors. The number of colors used in a minimal edgecoloring of a multigraph G is called the chromatic
Parallel algorithms for the edgecoloring and edgecoloring update problems
 J. Parallel Distributed Comput
, 1996
"... Let G(V, E) be a simple undirected graph with a maximum vertex degree �(G) (or � for short), �V � � n and �E � � m. An edgecoloring of G is an assignment to each edge in G a color such that all edges sharing a common vertex have different colors. The minimum number of colors needed is denoted by ..."
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by ��(G) (called the chromatic index). For a simple graph G, itis known that � � ��(G) � ��1. This paper studies two edgecoloring problems. The first problem is to perform edgecoloring for an existing edgecolored graph G with ��1 colors stemming from the addition of a new vertex into G. The proposed
EdgeColoring and fColoring for Various Classes of Graphs
 MATCH Commun. Math. Comput. Chem
, 1999
"... In an ordinary edgecoloring of a graph each color... This paper gives efficient sequential and parallel algorithms to find ordinary edgecolorings and fcolorings for various classes of graphs such as bipartite graphs, planar graphs, and graphs having fixed degeneracy, treewidth, genus, arboricity ..."
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In an ordinary edgecoloring of a graph each color... This paper gives efficient sequential and parallel algorithms to find ordinary edgecolorings and fcolorings for various classes of graphs such as bipartite graphs, planar graphs, and graphs having fixed degeneracy, treewidth, genus
Graphical models, exponential families, and variational inference
, 2008
"... The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fiel ..."
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likelihoods, marginal probabilities and most probable configurations. We describe how a wide varietyof algorithms — among them sumproduct, cluster variational methods, expectationpropagation, mean field methods, maxproduct and linear programming relaxation, as well as conic programming relaxations — can
Finding community structure in networks using the eigenvectors of matrices
, 2006
"... We consider the problem of detecting communities or modules in networks, groups of vertices with a higherthanaverage density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as “modularity ” over possible div ..."
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We consider the problem of detecting communities or modules in networks, groups of vertices with a higherthanaverage density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as “modularity ” over possible
Improved EdgeColoring Algorithms for Planar Graphs
 JOURNAL OF ALGORITHMS
, 1996
"... We consider the problem of edgecoloring planar graphs. It is known that a planar graph G with maximum degree \Delta \geq 8 can be colored with \Delta colors. We present two algorithms which find such a coloring when \Delta \geq 9. The first one is a sequential O(n log n) time algorithm. The other o ..."
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We consider the problem of edgecoloring planar graphs. It is known that a planar graph G with maximum degree \Delta \geq 8 can be colored with \Delta colors. We present two algorithms which find such a coloring when \Delta \geq 9. The first one is a sequential O(n log n) time algorithm. The other
Results 1  10
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7,375