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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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Cited by 41 (7 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.
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 (straight line grid) drawing of a gr ..."
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Cited by 26 (20 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 (straight line 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.
Nonrepetitive colorings of graphs of bounded treewidth, manuscript
, 2003
"... A sequence of the form s1s2...sms1s2...sm is called a repetition. A vertexcoloring of a graph is called nonrepetitive if none of its paths is repetitively colored. We answer a question of Grytczuk [5] by proving that every outerplanar graph has a nonrepetitive 12coloring. We also show that graphs ..."
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Cited by 13 (0 self)
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A sequence of the form s1s2...sms1s2...sm is called a repetition. A vertexcoloring of a graph is called nonrepetitive if none of its paths is repetitively colored. We answer a question of Grytczuk [5] by proving that every outerplanar graph has a nonrepetitive 12coloring. We also show that graphs of treewidth t have nonrepetitive 4 tcolorings. 1
New acyclic and star coloring algorithms with application to computing Hessians
 SIAM JOURNAL ON SCIENTIFIC COMPUTING VOL
, 2007
"... Acyclic and star coloring problems are specialized vertex coloring problems that arise in the efficient computation of Hessians using automatic differentiation or finite differencing, when both sparsity and symmetry are exploited. We present an algorithmic paradigm for finding heuristic solutions fo ..."
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Cited by 11 (7 self)
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Acyclic and star coloring problems are specialized vertex coloring problems that arise in the efficient computation of Hessians using automatic differentiation or finite differencing, when both sparsity and symmetry are exploited. We present an algorithmic paradigm for finding heuristic solutions for these two NPhard problems. The underlying common technique is the exploitation of the structure of twocolored induced subgraphs. For a graph G on n vertices and m edges, the time complexity of our star coloring algorithm is O(nd2), where dk, a generalization of vertex degree, denotes the average number of distinct paths of length at most k edges starting at a vertex in G. The time complexity of our acyclic coloring algorithm is larger by a multiplicative factor involving the inverse of Ackermann’s function. The space complexity of both algorithms is O(m). To the best of our knowledge, our work is the first practical algorithm for the acyclic coloring problem. For the star coloring problem, our algorithm uses fewer colors and is considerably faster than a previously known O(nd3)time algorithm. Computational results from experiments on various largesize test graphs demonstrate that the algorithms are fast and produce highly effective solutions. The use of these algorithms in Hessian computation is expected to reduce overall runtime drastically.
Graphs with maximum degree 5 are acyclically 7colorable
, 2011
"... An acyclic coloring is a proper coloring with the additional property that the union of any two color classes induces a forest. We show that every graph with maximum degree at most 5 has an acyclic 7coloring. We also show that every graph with maximum degree at most r has an acyclic (1 + ⌊ (r+1)2 4 ..."
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Cited by 2 (0 self)
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An acyclic coloring is a proper coloring with the additional property that the union of any two color classes induces a forest. We show that every graph with maximum degree at most 5 has an acyclic 7coloring. We also show that every graph with maximum degree at most r has an acyclic (1 + ⌊ (r+1)2 4 ⌋)coloring.
Acyclic Coloring of Graphs of Maximum Degree ∆
"... An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two neighbors in G are assigned the same color and (ii) no bicolored cycle can exist in G. The acyclic chromatic number of G is the least number of colors necessary to acyclically color G, and is denoted by a(G). We sho ..."
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Cited by 2 (0 self)
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An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two neighbors in G are assigned the same color and (ii) no bicolored cycle can exist in G. The acyclic chromatic number of G is the least number of colors necessary to acyclically color G, and is denoted by a(G). We show that any graph of maximum degree ∆ has acyclic chromatic number at most ∆(∆−1) 2 for any ∆ ≥ 5, and we give an O(n ∆ 2) algorithm to acyclically color any graph of maximum degree ∆ with the above mentioned number of colors. This result is roughly two times better than the best general upper bound known so far, yielding a(G) ≤ ∆( ∆ − 1) + 2 [ACK + 04]. By a deeper study of the case ∆ = 5, we also show that any graph of maximum degree 5 can be acyclically colored with at most 9 colors, and give a linear time algorithm to achieve this bound.
Star coloring of sparse graphs
"... A proper coloring of the vertices of a graph is called a star coloring if the union of every two color classes induce a star forest. The star chromatic number χs(G) is the smallest number of colors required to obtain a star coloring of G. In this paper, we study the relationship between the star chr ..."
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Cited by 1 (0 self)
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A proper coloring of the vertices of a graph is called a star coloring if the union of every two color classes induce a star forest. The star chromatic number χs(G) is the smallest number of colors required to obtain a star coloring of G. In this paper, we study the relationship between the star chromatic number χs(G) and the maximum average degree Mad(G) of a graph G. We prove that: 1. If G is a graph with Mad(G) < 26, then χs(G) ≤ 4.
DEGENERATE AND STAR COLORINGSOFGRAPHSON SURFACES
, 2008
"... We study the degenerate, the star and the degenerate star chromatic numbers and their relation to the genus of graphs. As a tool we prove the following strengthening of a result of Fertin et al. [8]: If G is a graph of maximum degree Δ, then G admits a degenerate star coloring using O(Δ 3/2) colors. ..."
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We study the degenerate, the star and the degenerate star chromatic numbers and their relation to the genus of graphs. As a tool we prove the following strengthening of a result of Fertin et al. [8]: If G is a graph of maximum degree Δ, then G admits a degenerate star coloring using O(Δ 3/2) colors. We use this result to prove that every graph of genus g admits a degenerate star coloring with O(g 3/5) colors. It is also shown that these results are sharp up to a logarithmic factor. Key words: graph coloring, degenerate coloring, acyclic coloring, star coloring, planar graph, genus
A Guide to the Discharging Method
, 2013
"... We provide a “howto” guide to the use and application of the Discharging Method. Our aim is not to exhaustively survey results that have been proved by this technique, but rather to demystify the technique and facilitate its wider use. Along the way, we present some new proofs and new problems. ..."
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We provide a “howto” guide to the use and application of the Discharging Method. Our aim is not to exhaustively survey results that have been proved by this technique, but rather to demystify the technique and facilitate its wider use. Along the way, we present some new proofs and new problems.