DOCSLIB.ORG

Parallel and On-Line Graph Coloring 1 Introduction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Parallel and On-Line Graph Coloring 1 Introduction Parallel and Online Graph Coloring Magns M Halldrsson Science Institute University of Iceland IS Reykjavik Iceland Abstract We discover a surprising connection b etween graph coloring in two orthogonal paradigms parallel and online computing We present a randomized online coloring algorithm with p a p erformance ratio of O n log n an improvement of log n factor over the previous b est known algorithm of Vishwanathan Also from the same principles we construct a parallel coloring algorithm with the same p erformance ratio for the rst such result As a byproduct we obtain a parallel approximation for the indep endent set problem Introduction At the heart of most partition and allo cation problems is a graph coloring question The graph coloring problem involves assigning values or colors to the vertices of a graph so that adjacent vertices are assigned distinct colors with the ob jective of minimizing the number of colors used We seek graph coloring algorithms that are b oth eective and ecient Eciency means among other things using computation time b ounded by a p olynomial in the size of the input Eectiveness means that the quality of the coloring is never much worse than the optimal coloring it is measured in terms of the performance ratio of the algorithm dened as the ratio of the number of colors used to the minimum number of colors needed maximized over all inputs In this pap er we present a reasonably eective and ecient graph coloring algorithm that yields b oth parallel and online algorithms A parallel algorithm is said to b e ecient if it uses time p olylogarithmic and pro cessors p olynomial in the size of the input The coloring problem is of central imp ortance in parallel and distributed computation as a means of scheduling nonconicting concurrent tasks Previous parallel algorithm have attained the degree b ound of Bro okss theorem coloring of planar graphs and some other sp ecial cases but no nontrivial p erformance guarantee on general k k graph was known We present a parallel algorithm that uses at most n colors on k colorable graphs for a p erformance ratio of O n log n where n is the number of vertices An algorithm is said to b e online if the color assigned to vertex i is indep endent of ver tices with index greater than i This corresp onds to various dynamic situations such as the assignments of frequencies to mobile users The online graph coloring problem has b een widely studied and for various other reasons as well In particular an algorithm that improves the trivial p erformance ratio to O n log n was given by Lovsz Saks and Trotter and p p n on colorable graph and O n log n ratio on a randomized algorithm with ratio of O email mmhrhihiis Research partly p erformed at Rutgers University Tokyo Institute of Technology and Japan Advanced Institute of Science and Technology Hokuriku and supp orted there by a DIMACS Graduate Fellowship a Toshiba Endowed Chair and a PFU Endowed Chair resp ectively general graphs was given by Vishwanathan We improve here the randomized algorithm of for a p erformance ratio of O n log n We rst present in Section a sequential graph coloring algorithm with a p erformance ratio of O n log n and use it to derive b oth a parallel algorithm in Section and a randomized on line algorithm in Section with the same p erformance ratio The randomized algorithm is an p log n improvement of an algorithm of Vishwanathan and the p erformance ratio of O n compares favorably with the lower b ound for randomized online coloring of n log n The p erformance ratio of the parallel algorithm is the rst of its kind It also compares favor ably with the b est p erformance ratio known for sequential coloring of O nlog log n log n and the b est announced lower b ound known for that quantity assuming P NP of n for each Finally using the parallel coloring algorithm we obtain in Section a O nlog log n log n p erformance ratio for parallel indep endent set problem This also com pares favorably to the sequential upp er b ound of O n log n and the announced lower b ound of n for each See Table for comparisons Online Parallel Sequential Deterministic Randomized p log n O nlg lg n log n Known upp er b ound O n log n O n Here O n log n O n log n Known lower b ounds n log n n log n n n Table New and known results on approximate graph coloring This connection b etween apparently unrelated algorithmic paradigms is quite surprising as the two paradigms we study are seemingly orthogonal and incompatible Parallel computation is inherently oline and online computation inherently sequential Nevertheless there are certain strong similarities in particular b oth require a lo ose coupling among subproblems Parallel algorithms b enet greatly from lo cality which can b e thought of as a case of limited information similar to the online regime Online algorithms know more ab out certain other subproblems but with less control over what they learn Approximate Coloring Algorithms In this section we present and analyze the underlying sequential oline coloring algorithm that forms the basis of the parallel and online variants Let us rst review what we have learned from previous metho ds First consider an algorithm due to Wigderson Given a threecolorable graph on n vertices we lo ok for a vertex of degree p n The neighborho o d of that vertex must b e twocolorable so we color these neighbors at least with two fresh new colors in linear time and remove them from the graph We rep eat this until p p n for at most n iterations The remaining graph can easily b e maximum degree falls b elow p p n colors for a combined total of n colors This principle colored using max degree can b e generalized to graphs of larger chromatic number where the key observation is that the neighborho o d of any no de in a k colorable graph must b e k colorable This allows us the break the problem into a collection of somewhat easier subproblems The problem with this approach is that it lls the color classes in sequence We need a metho d that accumulates many color classes concurrently rather than searching for individual indep endent sets We extract an idea from the online algorithm of Vishwanathan A simple greedy algorithm sometimes referred to as First Fit works as follows The vertices are pro cessed in sequence according to some arbitrary order and a vertex is assigned to the rst color to which none of the preceding neighbors of the vertex have b een assigned If we are given a limited number of colors it may not b e p ossible to color all the vertices The resulting coloring of a subset of the vertices is called a maximal partial coloring of the graph A maximal partial coloring is an assignment of some of the no des into xed number of greedy color classes so that each remaining no de is adjacent to some no de in each greedy class The imp ortant p oint is that each color class partitions the remaining no des into subsets of the neighborho o ds of the no des in the color class If we choose a small class the remaining uncolored no des induce relatively few subproblems The total number of colors used is then the number of greedy color classes plus the sum of the number of colors used in the subproblems Such a maximal partial coloring is easy to achieve sequentially via the First Fit algorithm which assigns a vertex to the rst compatible color class if one exists KColorGk b egin if k BipartiteColorG else if k log n Assign each vertex a dierent color else k k k s sn k nk ResidueNo des MaximalPartialColorGs Find the smallest greedy color class and let w w b e its no des p Partition the ResidueNo des into R R such that no des in R are adjacent to w p i i for i to p ColorR k i end Figure An approximation algorithm for k colorable graphs We state the main algorithm in Figure The algorithm nds a maximal partial coloring partitions the remaining no des around the smallest color class and recurses on those subprob lems with the recursion b ottoming out at bipartite graphs When the chromatic number is to o large for us to handle with resp ect to the size of the graph we settle on the trivial coloring of one color p er vertex Theorem The number of colors used by Color on k colorable graphs is at most f k log k k k n where f k O k Proof Let An k denote the maximum number of colors used by the algorithm to color any k k k k colorable graph on n vertices Let f k denote k k and let k k An k denote f k n Our prop osition is that An k An k for all p ositive integers n k k The pro of pro ceeds by induction on k with the base case k established by an optimal bipartite algorithm Hence assume from here on that An k An k for all p ositive integers n The number of colors used by our algorithm is the number of greedy colors s sn k plus the number of colors used in the t subproblems each on m vertices To maintain sanity j and simplicity of presentation we ignore all ceilings Note that An k k sn k so our main task is to show that the number of colors used in the subproblems b e at most k k k sn k k n Intuitively we nd that the worst case o ccurs when the subproblems are all of equal size and then derive the worstcase number of subproblems If g is the total number of vertices in the greedy color classes then the number of vertices in the smallest class and thus the number of subproblems is at most g s By the inductive assumption we can assume that the number of colors used in the subproblems is b ounded by A Thus g s X Am
Load more

Recommended publications
  • Forbidden Subgraph Characterization of Quasi-Line Graphs Medha Dhurandhar [email protected]
    Forbidden Subgraph Characterization of Quasi-line Graphs Medha Dhurandhar [email protected] Abstract: Here in particular, we give a characterization of Quasi-line Graphs in terms of forbidden induced subgraphs. In general, we prove a necessary and sufficient condition for a graph to be a union of two cliques. 1. Introduction: A graph is a quasi-line graph if for every vertex v, the set of neighbours of v is expressible as the union of two cliques. Such graphs are more general than line graphs, but less general than claw-free graphs. In [2] Chudnovsky and Seymour gave a constructive characterization of quasi-line graphs. An alternative characterization of quasi-line graphs is given in [3] stating that a graph has a fuzzy reconstruction iff it is a quasi-line graph and also in [4] using the concept of sums of Hoffman graphs. Here we characterize quasi-line graphs in terms of the forbidden induced subgraphs like line graphs. We consider in this paper only finite, simple, connected, undirected graphs. The vertex set of G is denoted by V(G), the edge set by E(G), the maximum degree of vertices in G by Δ(G), the maximum clique size by (G) and the chromatic number by G). N(u) denotes the neighbourhood of u and N(u) = N(u) + u. For further notation please refer to Harary [3]. 2. Main Result: Before proving the main result we prove some lemmas, which will be used later. Lemma 1: If G is {3K1, C5}-free, then either 1) G ~ K|V(G)| or 2) If v, w V(G) are s.t.
    [Show full text]
  • When the Vertex Coloring of a Graph Is an Edge Coloring of Its Line Graph — a Rare Coincidence
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of the Academy's Library When the vertex coloring of a graph is an edge coloring of its line graph | a rare coincidence Csilla Bujt¶as 1;¤ E. Sampathkumar 2 Zsolt Tuza 1;3 Charles Dominic 2 L. Pushpalatha 4 1 Department of Computer Science and Systems Technology, University of Pannonia, Veszpr¶em,Hungary 2 Department of Mathematics, University of Mysore, Mysore, India 3 Alfr¶edR¶enyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary 4 Department of Mathematics, Yuvaraja's College, Mysore, India Abstract The 3-consecutive vertex coloring number Ã3c(G) of a graph G is the maximum number of colors permitted in a coloring of the vertices of G such that the middle vertex of any path P3 ½ G has the same color as one of the ends of that P3. This coloring constraint exactly means that no P3 subgraph of G is properly colored in the classical sense. 0 The 3-consecutive edge coloring number Ã3c(G) is the maximum number of colors permitted in a coloring of the edges of G such that the middle edge of any sequence of three edges (in a path P4 or cycle C3) has the same color as one of the other two edges. For graphs G of minimum degree at least 2, denoting by L(G) the line graph of G, we prove that there is a bijection between the 3-consecutive vertex colorings of G and the 3-consecutive edge col- orings of L(G), which keeps the number of colors unchanged, too.
    [Show full text]
  • Counting Independent Sets in Graphs with Bounded Bipartite Pathwidth∗
    Counting independent sets in graphs with bounded bipartite pathwidth∗ Martin Dyery Catherine Greenhillz School of Computing School of Mathematics and Statistics University of Leeds UNSW Sydney, NSW 2052 Leeds LS2 9JT, UK Australia [email protected] [email protected] Haiko M¨uller∗ School of Computing University of Leeds Leeds LS2 9JT, UK [email protected] 7 August 2019 Abstract We show that a simple Markov chain, the Glauber dynamics, can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class. The class is determined by boundedness of a new graph parameter called bipartite pathwidth. This result, which we prove for the more general hardcore distribution with fugacity λ, can be viewed as a strong generalisation of Jerrum and Sinclair's work on approximately counting matchings, that is, independent sets in line graphs. The class of graphs with bounded bipartite pathwidth includes claw-free graphs, which generalise line graphs. We consider two further generalisations of claw-free graphs and prove that these classes have bounded bipartite pathwidth. We also show how to extend all our results to polynomially-bounded vertex weights. 1 Introduction There is a well-known bijection between matchings of a graph G and independent sets in the line graph of G. We will show that we can approximate the number of independent sets ∗A preliminary version of this paper appeared as [19]. yResearch supported by EPSRC grant EP/S016562/1 \Sampling in hereditary classes". zResearch supported by Australian Research Council grant DP190100977. 1 in graphs for which all bipartite induced subgraphs are well structured, in a sense that we will define precisely.
    [Show full text]
  • General Approach to Line Graphs of Graphs 1
    DEMONSTRATIO MATHEMATICA Vol. XVII! No 2 1985 Antoni Marczyk, Zdzislaw Skupien GENERAL APPROACH TO LINE GRAPHS OF GRAPHS 1. Introduction A unified approach to the notion of a line graph of general graphs is adopted and proofs of theorems announced in [6] are presented. Those theorems characterize five different types of line graphs. Both Krausz-type and forbidden induced sub- graph characterizations are provided. So far other authors introduced and dealt with single spe- cial notions of a line graph of graphs possibly belonging to a special subclass of graphs. In particular, the notion of a simple line graph of a simple graph is implied by a paper of Whitney (1932). Since then it has been repeatedly introduc- ed, rediscovered and generalized by many authors, among them are Krausz (1943), Izbicki (1960$ a special line graph of a general graph), Sabidussi (1961) a simple line graph of a loop-free graph), Menon (1967} adjoint graph of a general graph) and Schwartz (1969; interchange graph which coincides with our line graph defined below). In this paper we follow another way, originated in our previous work [6]. Namely, we distinguish special subclasses of general graphs and consider five different types of line graphs each of which is defined in a natural way. Note that a similar approach to the notion of a line graph of hypergraphs can be adopted. We consider here the following line graphsi line graphs, loop-free line graphs, simple line graphs, as well as augmented line graphs and augmented loop-free line graphs. - 447 - 2 A. Marczyk, Z.
    [Show full text]
  • Effective and Efficient Dynamic Graph Coloring
    Effective and Efficient Dynamic Graph Coloring Long Yuanx, Lu Qinz, Xuemin Linx, Lijun Changy, and Wenjie Zhangx x The University of New South Wales, Australia zCentre for Quantum Computation & Intelligent Systems, University of Technology, Sydney, Australia y The University of Sydney, Australia x{longyuan,lxue,zhangw}@cse.unsw.edu.au; [email protected]; [email protected] ABSTRACT (1) Nucleic Acid Sequence Design in Biochemical Networks. Given Graph coloring is a fundamental graph problem that is widely ap- a set of nucleic acids, a dependency graph is a graph in which each plied in a variety of applications. The aim of graph coloring is to vertex is a nucleotide and two vertices are connected if the two minimize the number of colors used to color the vertices in a graph nucleotides form a base pair in at least one of the nucleic acids. such that no two incident vertices have the same color. Existing The problem of finding a nucleic acid sequence that is compatible solutions for graph coloring mainly focus on computing a good col- with the set of nucleic acids can be modelled as a graph coloring oring for a static graph. However, since many real-world graphs are problem on a dependency graph [57]. highly dynamic, in this paper, we aim to incrementally maintain the (2) Air Traffic Flow Management. In air traffic flow management, graph coloring when the graph is dynamically updated. We target the air traffic flow can be considered as a graph in which each vertex on two goals: high effectiveness and high efficiency.
    [Show full text]
  • Degrees & Isomorphism: Chapter 11.1 – 11.4
    “mcs” — 2015/5/18 — 1:43 — page 393 — #401 11 Simple Graphs Simple graphs model relationships that are symmetric, meaning that the relationship is mutual. Examples of such mutual relationships are being married, speaking the same language, not speaking the same language, occurring during overlapping time intervals, or being connected by a conducting wire. They come up in all sorts of applications, including scheduling, constraint satisfaction, computer graphics, and communications, but we’ll start with an application designed to get your attention: we are going to make a professional inquiry into sexual behavior. Specifically, we’ll look at some data about who, on average, has more opposite-gender partners: men or women. Sexual demographics have been the subject of many studies. In one of the largest, researchers from the University of Chicago interviewed a random sample of 2500 people over several years to try to get an answer to this question. Their study, published in 1994 and entitled The Social Organization of Sexuality, found that men have on average 74% more opposite-gender partners than women. Other studies have found that the disparity is even larger. In particular, ABC News claimed that the average man has 20 partners over his lifetime, and the av- erage woman has 6, for a percentage disparity of 233%. The ABC News study, aired on Primetime Live in 2004, purported to be one of the most scientific ever done, with only a 2.5% margin of error. It was called “American Sex Survey: A peek between the sheets”—raising some questions about the seriousness of their reporting.
    [Show full text]
  • Discrete Mathematics Panconnected Index of Graphs
    Discrete Mathematics 340 (2017) 1092–1097 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Panconnected index of graphs Hao Li a,*, Hong-Jian Lai b, Yang Wu b, Shuzhen Zhuc a Department of Mathematics, Renmin University, Beijing, PR China b Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA c Department of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, PR China article info a b s t r a c t Article history: For a connected graph G not isomorphic to a path, a cycle or a K1;3, let pc(G) denote the Received 22 January 2016 smallest integer n such that the nth iterated line graph Ln(G) is panconnected. A path P is Received in revised form 25 October 2016 a divalent path of G if the internal vertices of P are of degree 2 in G. If every edge of P is a Accepted 25 October 2016 cut edge of G, then P is a bridge divalent path of G; if the two ends of P are of degree s and Available online 30 November 2016 t, respectively, then P is called a divalent (s; t)-path. Let `(G) D maxfm V G has a divalent g Keywords: path of length m that is not both of length 2 and in a K3 . We prove the following. Iterated line graphs (i) If G is a connected triangular graph, then L(G) is panconnected if and only if G is Panconnectedness essentially 3-edge-connected. Panconnected index of graphs (ii) pc(G) ≤ `(G) C 2.
    [Show full text]
  • THE CRITICAL GROUP of a LINE GRAPH: the BIPARTITE CASE Contents 1. Introduction 2 2. Preliminaries 2 2.1. the Graph Laplacian 2
    THE CRITICAL GROUP OF A LINE GRAPH: THE BIPARTITE CASE JOHN MACHACEK Abstract. The critical group K(G) of a graph G is a finite abelian group whose order is the number of spanning forests of the graph. Here we investigate the relationship between the critical group of a regular bipartite graph G and its line graph line G. The relationship between the two is known completely for regular nonbipartite graphs. We compute the critical group of a graph closely related to the complete bipartite graph and the critical group of its line graph. We also discuss general theory for the critical group of regular bipartite graphs. We close with various examples demonstrating what we have observed through experimentation. The problem of classifying the the relationship between K(G) and K(line G) for regular bipartite graphs remains open. Contents 1. Introduction 2 2. Preliminaries 2 2.1. The graph Laplacian 2 2.2. Theory of lattices 3 2.3. The line graph and edge subdivision graph 3 2.4. Circulant graphs 5 2.5. Smith normal form and matrices 6 3. Matrix reductions 6 4. Some specific regular bipartite graphs 9 4.1. The almost complete bipartite graph 9 4.2. Bipartite circulant graphs 11 5. A few general results 11 5.1. The quotient group 11 5.2. Perfect matchings 12 6. Looking forward 13 6.1. Odd primes 13 6.2. The prime 2 14 6.3. Example exact sequences 15 References 17 Date: December 14, 2011. A special thanks to Dr. Victor Reiner for his guidance and suggestions in completing this work.
    [Show full text]
  • The Cube of Every Connected Graph Is 1-Hamiltonian
    JOURNAL OF RESEARCH of the National Bureau of Standards- B. Mathematical Sciences Vol. 73B, No.1, January- March 1969 The Cube of Every Connected Graph is l-Hamiltonian * Gary Chartrand and S. F. Kapoor** (November 15,1968) Let G be any connected graph on 4 or more points. The graph G3 has as its point set that of C, and two distinct pointE U and v are adjacent in G3 if and only if the distance betwee n u and v in G is at most three. It is shown that not only is G" hamiltonian, but the removal of any point from G" still yields a hamiltonian graph. Key Words: C ube of a graph; graph; hamiltonian. Let G be a graph (finite, undirected, with no loops or multiple lines). A waLk of G is a finite alternating sequence of points and lines of G, beginning and ending with a point and where each line is incident with the points immediately preceding and followin g it. A walk in which no point is repeated is called a path; the Length of a path is the number of lines in it. A graph G is connected if between every pair of distinct points there exists a path, and for such a graph, the distance between two points u and v is defined as the length of the shortest path if u 0;1= v and zero if u = v. A walk with at least three points in which the first and last points are the same but all other points are dis tinct is called a cycle.
    [Show full text]
  • On a {K4,K2,2,2}-Ultrahomogeneous Graph
    On a {K4, K2,2,2}-ultrahomogeneous graph Italo J. Dejter University of Puerto Rico Rio Piedras, PR 00931-3355 [email protected] Abstract The existence of a connected 12-regular {K4, K2,2,2}-ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane taken as vertices. This graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of K4 and as the edge-disjoint union of 21 induced copies of K2,2,2, with no more copies of K4 or K2,2,2 existing in G. Moreover, each edge of G is shared by exactly one copy of K4 and one of K2,2,2. While the line graphs of d-cubes, (3 ≤ d ∈ ZZ), are {Kd, K2,2}-ultrahomogeneous, G is not even line-graphical. In addition, the chordless 6-cycles of G are seen to play an interesting role and some self-dual configurations associated to G with 2-arc-transitive, arc-transitive and semisymmetric Levi graphs are considered. 1 Introduction m Let H be a connected regular graph and let m, n ∈ ZZwith 1 <m<n. An {H}n -graph is a arXiv:0704.1493v3 [math.CO] 20 Oct 2008 connected graph that: (a) is representable as an edge-disjoint union of n induced copies of H; (b) has exactly m copies of H incident to each vertex, with no two such copies sharing more than one vertex; and (c) has exactly n copies of H as induced subgraphs isomorphic to H.
    [Show full text]
  • Application of Iterated Line Graphs to Biomolecular Conformation
    AN APPLICATION OF ITERATED LINE GRAPHS TO BIOMOLECULAR CONFORMATION DANIEL B. DIX Abstract. Graph theory has long been applied to molecular structure in re- gard to the covalent bonds between atoms. Here we extend the graph G whose vertices are atoms and whose edges are covalent bonds to allow a description of the conformation (or shape) of the molecule in three dimensional space. We define GZ-trees to be a certain class of tree subgraphs Γ of a graph AL2(G), which we call the amalgamated twice iterated line graph of G, and show that each such rooted GZ-tree (Γ;r) defines a well-behaved system of molecular internal coordinates, generalizing those known to chemists as Z-matrices. We prove that these coordinates are the most general type which give a diffeo- morphism of an explicitly determined and very large open subset of molecular configuration space onto the Cartesian product of the overall position and orientation manifold and the internal coordinate space. We give examples of labelled rooted GZ-trees, describing three dimensional (3D) molecular struc- tures, for three types of molecules important in biochemistry: amino acids, nucleotides, and glucose. Finally, some graph theoretical problems natural from the standpoint of molecular conformation are discussed. Contents 1. Introduction 2 2. Preliminaries 5 2.1. Space, Poses, Affine Symmetries 5 2.2. Amalgamated Iterated Line Graphs 7 2.3. Coordinatizing an Orbit Space 10 3. From Configuration to Coordinates 14 3.1. Conforming Pose Assignments 14 3.2. Defining Internal Coordinates 17 4. The Main Theorem 21 4.1. Z-trees, GZ-trees, Statement 21 4.2.
    [Show full text]
  • 20.1 Line Graphs 20.2 Edge-Coloring
    20.1 Line Graphs The line graph of G, written as L(G), is the simple graph whose vertices are the edges of G. Edges v; u 2 E(G), represented as vertices v; u 2 V (L(G)), have an edge between them in L(G) if they share a common endpoint w 2 V (G): f(v; w); (u; w)g 2 E(G). There is a relationship between problems involving edges in G and problems involving the vertices in L(G): 1. An Eulerian Circuit in G is a spanning cycle in L(G). 2. A matching in G is an independent set in L(G). 3. A cut edge e = (u; v) in G is a cut vertex in L(G) if d(u); d(v) > 1. 4. Edge-coloring in G is equivalent to vertex coloring in L(G). Consider some H such that L(H) = G, and, while we aren't going to discuss it today, we can find such an H for G in polynomial time if it exists. An interesting notion is that we can theoretically exploit this relationship. Note that it's possible to compute a maximum matching in polynomial time in general on H while a maximum independent set requires exponential time in general on G; however there is a 1-to-1 correspondence between the results. Likewise, the same might be said of finding an optimal vertex coloring in general takes exponential time while an optimal edge coloring only requires polynomial time. We going to focus on the last problem today.
    [Show full text]