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134
What color is your Jacobian? Graph coloring for computing derivatives
 SIAM REV
, 2005
"... Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specific ..."
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Cited by 59 (11 self)
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Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specifics of the computational techniques employed. We consider eight variant vertexcoloring problems here. This article begins with a gentle introduction to the problem of computing a sparse Jacobian, followed by an overview of the historical development of the research area. Then we present a unifying framework for the graph models of the variant matrixestimation problems. The framework is based upon the viewpoint that a partition of a matrixinto structurally orthogonal groups of columns corresponds to distance2 coloring an appropriate graph representation. The unified framework helps integrate earlier work and leads to fresh insights; enables the design of more efficient algorithms for many problems; leads to new algorithms for others; and eases the task of building graph models for new problems. We report computational results on two of the coloring problems to support our claims. Most of the methods for these problems treat a column or a row of a matrixas an atomic entity, and partition the columns or rows (or both). A brief review of methods that do not fit these criteria is provided. We also discuss results in discrete mathematics and theoretical computer science that intersect with the topics considered here.
Good and Semistrong Colorings of Oriented Planar Graphs
 INF. PROCESSING LETTERS 51
, 1994
"... A kcoloring of an oriented graph G = (V, A) is an assignment c of one of the colors 1; 2; : : : ; k to each vertex of the graph such that, for every arc (x; y) of G, c(x) 6= c(y). The kcoloring is good if for every arc (x; y) of G there is no arc (z; t) 2 A such that c(x) = c(t) and c(y) = c(z). ..."
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Cited by 53 (21 self)
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A kcoloring of an oriented graph G = (V, A) is an assignment c of one of the colors 1; 2; : : : ; k to each vertex of the graph such that, for every arc (x; y) of G, c(x) 6= c(y). The kcoloring is good if for every arc (x; y) of G there is no arc (z; t) 2 A such that c(x) = c(t) and c(y) = c(z). A kcoloring is said to be semistrong if for every vertex x of G, c(z) 6= c(t) for any pair fz; tg of vertices of N \Gamma (x). We show that every oriented planar graph has a good coloring using at most 5 \Theta 2 4 colors and that every oriented planar graph G = (V; A) with d \Gamma (x) 3 for every x 2 V has a good and semistrong coloring using at most 4 \Theta 5 \Theta 2 4 colors.
Star Coloring of Graphs
, 2001
"... A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two neighbors are assigned the same color) such that any path of length 3 in G is not bicolored. The star ..."
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Cited by 42 (1 self)
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A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two neighbors are assigned the same color) such that any path of length 3 in G is not bicolored. The star
On the Maximum Average Degree and the Oriented Chromatic Number of a Graph
 Discrete Math
, 1995
"... The oriented chromatic number o(H) of an oriented graph H is defined as the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . The oriented chromatic number o(G) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In ..."
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Cited by 31 (15 self)
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The oriented chromatic number o(H) of an oriented graph H is defined as the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . The oriented chromatic number o(G) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In this paper we study the links between o(G) and mad(G) defined as the maximum average degree of the subgraphs of G. 1 Introduction and statement of results For every graph G we denote by V (G), with vG = jV (G)j, its set of vertices and by E(G), with e G = jE(G)j, its set of arcs or edges. A homomorphism from a graph G to a graph On leave of absence from the Institute of Mathematics, Novosibirsk, 630090, Russia. With support from Engineering and Physical Sciences Research Council, UK, grant GR/K00561, and from the International Science Foundation, grant NQ4000. y This work was partially supported by the Network DIMANET of the European Union and by the grant 960101614 of the Russian F...
Layout of Graphs with Bounded TreeWidth
 2002, SUBMITTED. STACKS, QUEUES AND TRACKS: LAYOUTS OF GRAPH SUBDIVISIONS 41
, 2004
"... A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queuenumber. A threedimensional (straightline grid) drawing of a gra ..."
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Cited by 31 (23 self)
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A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queuenumber. A threedimensional (straightline grid) drawing of a graph represents the vertices by points in Z and the edges by noncrossing linesegments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of threedimensional drawing of a graph G is closely related to the queuenumber of G. In particular, if G is an nvertex member of a proper minorclosed family of graphs (such as a planar graph), then G has a O(1) O(1) O(n) drawing if and only if G has O(1) queuenumber.
Oriented Graph Coloring
 DISCRETE MATH
, 2001
"... An oriented kcoloring of an oriented graph G (that is a digraph with no cycle of length 2) is a partition of its vertex set into k subsets such that (i) no two adjacent vertices belong to the same subset and (ii) all the arcs between any two subsets have the same direction. We survey the main resu ..."
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Cited by 24 (5 self)
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An oriented kcoloring of an oriented graph G (that is a digraph with no cycle of length 2) is a partition of its vertex set into k subsets such that (i) no two adjacent vertices belong to the same subset and (ii) all the arcs between any two subsets have the same direction. We survey the main results that have been obtained on oriented graph colorings.
PrimalDual Approximation Algorithms for Feedback Problems in Planar Graphs
 IPCO '96
, 1996
"... Given a subset of cycles of a graph, we consider the problem of finding a minimumweight set of vertices that meets all cycles in the subset. This problem generalizes a number of problems, including the minimumweight feedback vertex set problem in both directed and undirected graphs, the subset fee ..."
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Cited by 23 (3 self)
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Given a subset of cycles of a graph, we consider the problem of finding a minimumweight set of vertices that meets all cycles in the subset. This problem generalizes a number of problems, including the minimumweight feedback vertex set problem in both directed and undirected graphs, the subset feedback vertex set problem, and the graph bipartization problem, in which one must remove a minimumweight set of vertices so that the remaining graph is bipartite. We give a 9/4approximation algorithm for the general problem in planar graphs, given that the subset of cycles obeys certain properties. This results in 9/4approximation algorithms for the aforementioned feedback and bipartization problems in planar graphs. Our algorithms use the primaldual method for approximation algorithms as given in Goemans and Williamson [16]. We also show that our results have an interesting bearing on a conjecture of Akiyama and Watanabe [2] on the cardinality of feedback vertex sets in planar graphs.
The total chromatic number of any multigraph with maximum degree five is at most seven
, 1996
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