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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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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.
Acyclic colourings of planar graphs with large girth
 J. London Math. Soc
, 1999
"... A proper vertexcolouring of a graph is acyclic if there are no 2coloured cycles. It is known that every planar graph is acyclically 5colourable, and that there are planar graphs with acyclic chromatic number χ a � 5 and girth g � 4. It is proved here that a planar graph satisfies χ ..."
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A proper vertexcolouring of a graph is acyclic if there are no 2coloured cycles. It is known that every planar graph is acyclically 5colourable, and that there are planar graphs with acyclic chromatic number χ a � 5 and girth g � 4. It is proved here that a planar graph satisfies χ
Acyclic coloring of graphs with maximum degree five, http: //hal.archivesouvertes.fr/hal00375166/en
"... An acyclic kcoloring of a graph G is a proper vertex coloring of G which uses at most k colors such that the graph induced by the union of every two color classes is a forest. In this paper, we mainly prove that every 5connected graph with maximum degree five is acyclically 8colorable, improving ..."
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An acyclic kcoloring of a graph G is a proper vertex coloring of G which uses at most k colors such that the graph induced by the union of every two color classes is a forest. In this paper, we mainly prove that every 5connected graph with maximum degree five is acyclically 8colorable, improving partially [5]. hal00375166, version 1 8 Feb 2010 1