If the graph really is three-colorable, the provers should never report the same color.
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Arguments based on U.S. competitiveness are at least colorable in that scenario, if not entirely coherent.
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The adverse events at Alere are enough to make a colorable claim for a MAC, certainly.
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So long it's a colorable argument, they make it, it obviously on behalf of their client.
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"We shouldn't assume that untested therefore means colorable," said Steve Vladeck, a University of Texas law professor.
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The agency instead asks if there is a colorable interpretation that will support the policy result that the agency wants to reach.
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Prior to the most recent hearing, Judge Kavanaugh's record of significant judicial public service presented a colorable case for a Supreme Court appointment.
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He was later able to shrink the graph to 1,581 vertices and do a computer check to verify that it was not four-colorable.
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If we were in federal court, litigating criminal charges against the president, I think the "Trump is just Trump" defense would be colorable and tricky to overcome.
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The problem is easy to understand and start working on, and there is a clear measure of success: lowering the number of vertices in a non-four-colorable graph.
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"The judge today found that the allegations in our lawsuit 'are sufficient to state a colorable claim for breach of fiduciary duty against Ms. Redstone and NAI as CBS's controlling stockholder,'" the statement said.
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"I take no position as to Mr. Cooper's guilt or innocence at this time, but colorable factual questions have been raised about whether advances in DNA technology warrant limited retesting of certain physical evidence in this case," Brown wrote in the order.
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If the program does in fact halt, the provers should be able to win this game 100 percent of the time—similar to how if a graph is actually three-colorable, entangled provers should never report the same color for two connected vertices.
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If the testimony involved classified information, there might be a colorable argument if the former employee agreed the conversation was confidential, and both thought it was covered by executive privilege and held the conversation on that basis, but absent even that dubious argument, conversations with the president of the United States do not give him the power to revoke the First Amendment.
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A complete graph is uniquely colorable, because the only proper coloring is one that assigns each vertex a different color. Every k-tree is uniquely (k + 1)-colorable. The uniquely 4-colorable planar graphs are known to be exactly the Apollonian networks, that is, the planar 3-trees .
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In other words, if G × H is 2-colorable, then at least one of G and H must be 2-colorable as well. The next case has been proved long after the conjecture's statement, by : if the product G × H is 3-colorable, then one of G or H must also be 3-colorable. In particular, the conjecture is true whenever G or H is 4-colorable (since then the inequality χ(G × H) ≤ min {χ(G), χ(H)} can only be strict when G × H is 3-colorable). In the remaining cases, both graphs in the tensor product are at least 5-chromatic and progress has only been made for very restricted situations.
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However, there exist uniquely 3-edge-colorable graphs that do not fit into this classification, such as the graph of the triangular pyramid. If a cubic graph is uniquely 3-edge-colorable, it must have exactly three Hamiltonian cycles, formed by the edges with two of its three colors, but some cubic graphs with only three Hamiltonian cycles are not uniquely 3-edge-colorable . Every simple planar cubic graph that is uniquely 3-edge-colorable contains a triangle , but observed that the generalized Petersen graph G(9,2) is non- planar, triangle-free, and uniquely 3-edge-colorable. For many years it was the only known such graph, and it had been conjectured to be the only such graph (see and ) but now infinitely many triangle-free non-planar cubic uniquely 3-edge-colorable graphs are known .
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Defective coloring was introduced nearly simultaneously by Burr and Jacobson, Harary and Jones and Cowen, Cowen and Woodall. Surveys of this and related colorings are given by Marietjie Frick. Cowen, Cowen and Woodall focused on graphs embedded on surfaces and gave a complete characterization of all k and d such that every planar is (k, d)-colorable. Namely, there does not exist a d such that every planar graph is (1, d)- or (2, d)-colorable; there exist planar graphs which are not (3, 1)-colorable, but every planar graph is (3, 2)-colorable.
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A uniquely total colorable graph is a k-total-chromatic graph that has only one possible (proper) k-total-coloring up to permutation of the colors. Empty graphs, paths, and cycles of length divisible by 3 are uniquely total colorable graphs. conjectured that they are also the only members in this family. Some properties of a uniquely k-total-colorable graph G with n vertices: # χ″(G) = Δ(G) + 1 unless G = K2.
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Any graph with a nonempty set of edges requires at least two colors; if G and H are not 1-colorable, that is, they both contain an edge, then their product also contains an edge, and is hence not 1-colorable either. In particular, the conjecture is true when G or H is a bipartite graph, since then its chromatic number is either 1 or 2. Similarly, if two graphs G and H are not 2-colorable, that is, not bipartite, then both contain a cycle of odd length. Since the product of two odd cycle graphs contains an odd cycle, the product G × H is not 2-colorable either.
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The unique 3-edge-coloring of the generalized Petersen graph G(9,2) A uniquely edge-colorable graph is a k-edge-chromatic graph that has only one possible (proper) k-edge-coloring up to permutation of the colors. The only uniquely 2-edge-colorable graphs are the paths and the cycles. For any k, the stars K1,k are the only uniquely k-edge-colorable graphs. Moreover, conjectured and proved that, when k ≥ 4, they are also the only members in this family.
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Let H be the hypergraph: > { {1,2} , {3,4} , {1,2,3,4} } it is 2-colorable and remains 2-colorable upon removing any number of vertices from it. However, It is not bipartite, since to have exactly one green vertex in each of the first two hyperedges, we must have two green vertices in the last hyperedge.
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It can be shown that this does not depend on the choice of gallery connecting C_0 and C_n. Now, a building is a simplicial complex that is organized into apartments, each of which is a Coxeter complex (satisfying some coherence axioms). Buildings are colorable, since the Coxeter complexes that make them up are colorable.
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In particular, G is k-colorable if and only if it is Kk- colorable. If there are two homomorphisms G → H and H → Kk, then their composition G → Kk is also a homomorphism. In other words, if a graph H can be colored with k colors, and there is a homomorphism from G to H, then G can also be k-colored. Therefore, G → H implies χ(G) ≤ χ(H), where χ denotes the chromatic number of a graph (the least k for which it is k-colorable).
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More precisely, the problem is polynomial-time if the graph is 2-colorable, that is, it is bipartite, and is NP-complete otherwise.
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Therefore, a graph is k-colorable if and only if it has an orientation that admits no homomorphism from k+1. This statement can be strengthened slightly to say that a graph is k-colorable if and only if some orientation contains no directed path of length k (no k+1 as a subgraph). This is the Gallai–Hasse–Roy–Vitaver theorem.
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Every Apollonian network is also a uniquely 4-colorable graph. Because it is a planar graph, the four color theorem implies that it has a graph coloring with only four colors, but once the three colors of the initial triangle are selected, there is only one possible choice for the color of each successive vertex, so up to permutation of the set of colors it has exactly one 4-coloring. It is more difficult to prove, but also true, that every uniquely 4-colorable planar graph is an Apollonian network. Therefore, Apollonian networks may also be characterized as the uniquely 4-colorable planar graphs.
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A hypergraph is called bipartite if its vertex set V can be partitioned into two sets, X and Y, such that each hyperedge contains exactly one element of X. To see that this sense is stronger than 2-colorability, let H be a hypergraph on the vertices {1, 2, 3, 4} with the following hyperedges: > { {1,2,3} , {1,2,4} , {1,3,4} , {2,3,4} } This H is 2-colorable, for example by the partition X = {1,2} and Y = {3,4}. However, it is not exactly-2-colorable, since every set X with one element has an empty intersection with one hyperedge, and every set X with two or more elements has an intersection of size 2 or more with at least two hyperedges. Every bipartite graph G = (X+Y, E) is exactly-2-colorable. Hall's marriage theorem has been generalized from bipartite graphs to exactly-2-colorable hypergraphs; see Hall-type theorems for hypergraphs.
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A minimal imperfect graph is a graph in which every subgraph is perfect. The deletion of any vertex from a minimal imperfect graph leaves a uniquely colorable subgraph.
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There are other natural generalizations of bipartite graphs. A hypergraph is called balanced if it is essentially 2-colorable, and remains essentially 2-colorable upon deleting any number of vertices (see Balanced hypergraph). The properties of bipartiteness and balance do not imply each other. Bipartiteness does not imply balance. For example, let H be the hypergraph with vertices {1,2,3,4} and edges: > { {1,2,3} , {1,2,4} , {1,3,4} } It is bipartite by the partition X={1}, Y={2,3,4}.
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A graph that can be assigned a (proper) k-coloring is k-colorable, and it is k-chromatic if its chromatic number is exactly k. A subset of vertices assigned to the same color is called a color class, every such class forms an independent set. Thus, a k-coloring is the same as a partition of the vertex set into k independent sets, and the terms k-partite and k-colorable have the same meaning.
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Balance does not imply bipartiteness. Let H be the hypergraph: > { {1,2} , {3,4} , {1,2,3,4} } it is 2-colorable and remains 2-colorable upon removing any number of vertices from it. However, It is not bipartite, since to have exactly one green vertex in each of the first two hyperedges, we must have two green vertices in the last hyperedge. Bipartiteness does not imply balance. For example, let H be the hypergraph with vertices {1,2,3,4} and edges: > { {1,2,3} , {1,2,4} , {1,3,4} } It is bipartite by the partition X={1}, Y={2,3,4}.
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Summarized Conclusions While the business decisions that brought about the crisis were largely within the realm of acceptable business judgement, the actions to manipulate financial statements do give rise to "colorable claims", especially against the CEO and CFOs but also against the auditors. In the opinion of the Examiner, "colorable" is generally meant to mean that sufficient evidence exists to support legal action and possible recovery of losses. Repo 105 was not inherently improper, but its use here violated accounting principles that require all legitimate transactions to have a business purpose. Repo 105 solely existed to manipulate financial information.
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Together with the (4, 0)-coloring implied by the four color theorem, this solves defective chromatic number for the plane. Poh and Goddard showed that any planar graph has a special (3,2)-coloring in which each color class is a linear forest, and this can be obtained from a more general result of Woodall. For general surfaces, it was shown that for each genus g \geq 0, there exists a k=k(g) such that every graph on the surface of genus g is (4, k)-colorable. This was improved to (3, k)-colorable by Dan Archdeacon.
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The theorem is named after German mathematician Herbert Grötzsch, who published its proof in 1959. Grötzsch's original proof was complex. attempted to simplify it but his proof was erroneous.. In 2003, Carsten Thomassen derived an alternative proof from another related theorem: every planar graph with girth at least five is 3-list-colorable. However, Grötzsch's theorem itself does not extend from coloring to list coloring: there exist triangle-free planar graphs that are not 3-list-colorable.. In 1989, Richard Steinberg and Dan Younger gave the first correct proof, in English, of the dual version of this theorem.
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A slightly more general result is true: if a planar graph has at most three triangles then it is 3-colorable. However, the planar complete graph K4, and infinitely many other planar graphs containing K4, contain four triangles and are not 3-colorable. In 2009, Dvořák, Kráľ, and Thomas announced a proof of another generalization, conjectured in 1969 by L. Havel: there exists a constant d such that, if a planar graph has no two triangles within distance d of each other, then it can be colored with three colors.. This work formed part of the basis for Dvořák's 2015 European Prize in Combinatorics.. The theorem cannot be generalized to all nonplanar triangle-free graphs: not every nonplanar triangle-free graph is 3-colorable. In particular, the Grötzsch graph and the Chvátal graph are triangle-free graphs requiring four colors, and the Mycielskian is a transformation of graphs that can be used to construct triangle-free graphs that require arbitrarily high numbers of colors.
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The defendant was barred from using the Planned Parenthood mark, making "colorable" imitations, representing the defendant (by offering information, products, or services), and from taking any other action in creating confusion for internet users or consumers in relation to the Planned Parenthood mark.
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A hypergraph H = (V, E) is called 2-colorable if its vertex set V can be partitioned into two sets, X and Y, such that each hyperedge meets both X and Y. Equivalently, the vertices of H can be 2-colored so that no hyperedge is monochromatic. Every bipartite graph G = (X+Y, E) is 2-colorable: each edge contains exactly one vertex of X and one vertex of Y, so e.g. X can be colored blue and Y can be colored yellow and no edge is monochromatic. The property fo 2-colorability was first introduced by Felix Bernstein in the context of set families;.
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It is also NP- hard to color a 3-colorable graph with 4 colors and a k-colorable graph with k(log k ) / 25 colors for sufficiently large constant k. Computing the coefficients of the chromatic polynomial is #P-hard. In fact, even computing the value of \chi(G,k) is #P-hard at any rational point k except for k = 1 and k = 2. There is no FPRAS for evaluating the chromatic polynomial at any rational point k ≥ 1.5 except for k = 2 unless NP = RP. For edge coloring, the proof of Vizing’s result gives an algorithm that uses at most Δ+1 colors.
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It is exactly-2-colorable by the partition X = {1} and Y = {2,3,4}. However, it is not 3-partite: in every partition of V into 3 subsets, at least one subset contains two vertices, and thus at least one hyperedge contains two vertices from this subset.
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A bipartite graph is (a, b)-biregular if everyvertex in one part has degree a and every vertex in the other part has degree b. It has been conjectured that all such graphs have interval colorings. Hansen proved that every bipartite graph G with ∆(G) ≤ 3 is interval colorable.
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In graph theory, a uniquely colorable graph is a k-chromatic graph that has only one possible (proper) k-coloring up to permutation of the colors. Equivalently, there is only one way to partition its vertices into k independent sets and there is no way to partition them into k−1 independent sets.
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The following discussion is a summary based on the introduction to Every Planar Map is Four Colorable . Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it. The explanation here is reworded in terms of the modern graph theory formulation above. Kempe's argument goes as follows.
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Besides balance, there are alternative generalizations of bipartite graphs. A hypergraph is called bipartite if its vertex set V can be partitioned into two sets, X and Y, such that each hyperedge contains exactly one element of X (see bipartite hypergraph). Obviously every bipartite graph is 2-colorable. The properties of bipartiteness and balance do not imply each other.
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According to , word-representable graphs are relevant to various fields, thus providing a motivation to study the graphs. These fields are algebra, graph theory, computer science, combinatorics on words, and scheduling. Word-representable graphs are especially important in graph theory, since they generalise several important classes of graphs, e.g. circle graphs, 3-colorable graphs and comparability graphs.
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The concept of consecutive edge-coloring was introduced with the terminology 'interval edge coloring' by Asratian and Kamalian in 1987 in their paper "Interval colorings of edges of a multigraph". Since interval edge coloring of graphs was introduced mathematicians have been investigating the existence of interval edge colorable graphs as not all graphs allow interval edge coloring. A simple family of graphs that allows interval edge coloring is complete graph of even order and a counter example of family of graphs includes complete graphs of odd order. The smallest graph that doesnot allow interval colorability.There are even graphs discovered with 28 vertices and maximum degree 21 that is not interval colorable by Sevast’janov though the interval colorability of graphs with maximum degree lying between four and twelve is still unknown.
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More generally, for a function f assigning a positive integer f(v) to each vertex v, a graph G is f-choosable (or f-list-colorable) if it has a list coloring no matter how one assigns a list of f(v) colors to each vertex v. In particular, if f(v) = k for all vertices v, f-choosability corresponds to k-choosability.
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They responded that the rumors were due to a "misinterpretation of [Schmidt's] results" and obliged with a detailed article . Their magnum opus, Every Planar Map is Four- Colorable, a book claiming a complete and detailed proof (with a microfiche supplement of over 400 pages), appeared in 1989; it explained and corrected the error discovered by Schmidt as well as several further errors found by others .
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2017) ("The fact that ... proceeding ended a decade ago should not preclude him from obtaining relief under Rule 60(b) if the court concludes that he has raised a colorable claim that he meets this threshold actual-innocence standard ...."); Herring v. United States, 424 F.3d 384, 386-87 (3d Cir. 2005); Luna v. Bell, 887 F.3d 290, 294 (6th Cir. 2018); United States v.
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The Grötzsch graph is a triangle-free graph that cannot be colored with fewer than four colors Much research about triangle-free graphs has focused on graph coloring. Every bipartite graph (that is, every 2-colorable graph) is triangle-free, and Grötzsch's theorem states that every triangle-free planar graph may be 3-colored.; ). However, nonplanar triangle-free graphs may require many more than three colors.
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If G contains a clique of size k, then at least k colors are needed to color that clique; in other words, the chromatic number is at least the clique number: : \chi(G) \ge \omega(G). For perfect graphs this bound is tight. Finding cliques is known as the clique problem. The 2-colorable graphs are exactly the bipartite graphs, including trees and forests.
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Even though the Frankl–Rödl graphs with this parameter value have high chromatic number, semidefinite programming is unable to distinguish them from 3-colorable graphs.... However, for these graphs, algebraic methods based on polynomial sums of squares can provide stronger bounds that certify their need for many colors. This result challenges the optimality of semidefinite programming and the correctness of the unique games conjecture.
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Byrd, the next time the Court had considered arbitration in a securities dispute, decided as the McMahons were filing their lawsuit. But Justice Byron White, who had joined William O. Douglas's Scherk dissent (which had not considered the "colorable argument") had devoted a short concurrence with the unanimous opinion to noting that it was still an open question.Dean Witter Reynolds Inc. v. Byrd, , 224–25, White, J., concurring.
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A graph that can be assigned a -edge-coloring is said to be -edge-colorable. The smallest number of colors needed in a (proper) edge coloring of a graph is the chromatic index, or edge chromatic number, . The chromatic index is also sometimes written using the notation ; in this notation, the subscript one indicates that edges are one-dimensional objects. A graph is -edge-chromatic if its chromatic index is exactly .
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The definition of a word-representable graph works both in labelled and unlabelled cases since any labelling of a graph is equivalent to any other labelling. Also, the class of word-representable graphs is hereditary. Word-representable graphs generalise several important classes of graphs such as circle graphs, 3-colorable graphs and comparability graphs. Various generalisations of the theory of word- representable graphs accommodate representation of any graph.
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By joining the single arrows together and the double arrows together, one obtains a torus with seven mutually touching regions; therefore seven colors are necessary This construction shows the torus divided into the maximum of seven regions, each one of which touches every other. The four-color theorem applies not only to finite planar graphs, but also to infinite graphs that can be drawn without crossings in the plane, and even more generally to infinite graphs (possibly with an uncountable number of vertices) for which every finite subgraph is planar. To prove this, one can combine a proof of the theorem for finite planar graphs with the De Bruijn–Erdős theorem stating that, if every finite subgraph of an infinite graph is k-colorable, then the whole graph is also k-colorable . This can also be seen as an immediate consequence of Kurt Gödel's compactness theorem for first-order logic, simply by expressing the colorability of an infinite graph with a set of logical formulae.
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He dismissed a suggestion that a law meant to protect small investors need not be held applicable to transactions between sophisticated companies, pointing out that Alberto-Culver had small shareholders as well who might have been victimized and "the rules when the giants play are the same as when the pygmies enter the market."Scherk II, at 525–526, Douglas, J., dissenting. As to Stewart's "colorable argument", Douglas reiterated Wilko's wariness towards arbitration.
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As an immediate corollary of the last statement, one has that the recognition problem of word-representability is in NP. In 2014, Vincent Limouzy observed that it is an NP-complete problem to recognise whether a given graph is word-representable . Another important corollary to the key theorem is that any 3-colorable graph is word- representable. The last fact implies that many classical graph problems are NP-hard on word-representable graphs.
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"One would have thought that, after these amendments, the matter of Wilko's extension to Exchange Act claims at last would be uncontroversial," he wrote. "Yet, like a ghost reluctant to accept its eternal rest, the 'colorable argument' surfaced again" in White's Byrd concurrence.Shearson, at 243–48. He focused his criticism on the Court's reading of Wilko: Blackmun agreed with the majority that a possible exemption to the FAA must be supported by a finding of Congressional intent.
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Edge coloring Tietze's graph requires four colors; that is, its chromatic index is 4. Equivalently, the edges of Tietze's graph can be partitioned into four matchings, but no fewer. Tietze's graph matches part of the definition of a snark: it is a cubic bridgeless graph that is not 3-edge- colorable. However, some authors restrict snarks to graphs without 3-cycles and 4-cycles, and under this more restrictive definition Tietze's graph is not a snark.
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However, the same definitions apply to directed graphs and a directed graph is also equivalent to a unique core. Every graph and every directed graph contains its core as a retract and as an induced subgraph. For example, all complete graphs Kn and all odd cycles (cycle graphs of odd length) are cores. Every 3-colorable graph G that contains a triangle (that is, has the complete graph K3 as a subgraph) is homomorphically equivalent to K3.
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They have Hamiltonian cycles if and only if they are biconnected, in which case the outer face forms the unique Hamiltonian cycle. Every outerplanar graph is 3-colorable, and has degeneracy and treewidth at most 2\. The outerplanar graphs are a subset of the planar graphs, the subgraphs of series-parallel graphs, and the circle graphs. The maximal outerplanar graphs, those to which no more edges can be added while preserving outerplanarity, are also chordal graphs and visibility graphs.
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An Apollonian network The Goldner–Harary graph, a non-Hamiltonian Apollonian network In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction.
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Two greedy colorings of the same graph using different vertex orders. The right example generalizes to 2-colorable graphs with n vertices, where the greedy algorithm expends n/2 colors. The greedy algorithm considers the vertices in a specific order v_1,…, v_n and assigns to v_i the smallest available color not used by v_i’s neighbours among v_1,…, v_{i-1}, adding a fresh color if needed. The quality of the resulting coloring depends on the chosen ordering.
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But is not balanced. For example, if vertex 1 is removed, we get the restriction of H to {2,3,4}, which has the following hyperedges: > { {2,3} , {2,4} , {3,4} } It is not 2-colorable, since in any 2-coloring there are at least two vertices with the same color, and thus at least one of the hyperedges is monochromatic. Another way to see that H is not balanced is that it contains the odd-length cycle C = (2 - {1,2,3} - 3 - {1,3,4} - 4 - {1,2,4} - 2), and no edge of C contains all three vertices 2,3,4 of C. Tripartiteness does not imply balance. For example, let H be the tripartite hypergraph with vertices {1,2},{3,4},{5,6} and edges: > { {1,3,5}, {2,4,5}, {1,4,6} } It is not balanced since if we remove the vertices 2,3,6, the remainder is: > { {1,5}, {4,5}, {1,4} } which is not colorable since it is a 3-cycle. Another way to see that it is not balanced is that It contains the odd-length cycle C = (1 - {1,3,5} - 5 - {2,4,5} - 4 - {1,4,6} - 1), and no edge of C contains all three vertices 1,4,5 of C.
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U.S. District Court Judge Harold P. Burke agreed with the complaint. His instructions were that Bully Hill was enjoined from "using the word Taylor or any colorable imitation thereof in connection with any labeling, packaging materials, advertising, or promotional material for any of defendant's products." The Taylor name was then blotted out wherever it appeared on Bully Hill products. During the court battle, sales at Bully Hill Vineyards climbed from $650,000 in 1977 to more than $2 million in 1980.
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The original proof of the De Bruijn–Erdős theorem, by De Bruijn, used transfinite induction. provided the following very short proof, based on Tychonoff's compactness theorem in topology. Suppose that, for the given infinite graph , every finite subgraph is -colorable, and let be the space of all assignments of the colors to the vertices of (regardless of whether they form a valid coloring). Then may be given a topology as a product space , where denotes the set of vertices of the graph.
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As with classical colorings, the reverse implication always holds: if G (or H, symmetrically) is K-colorable, then G × H is easily K-colored by using the same values independently of H. Hedetniemi's conjecture is then equivalent to the statement that each complete graph is multiplicative. The above known cases are equivalent to saying that K1, K2, and K3 are multiplicative. The case of K4 is widely open. On the other hand, the proof of has been generalized by to show that all cycle graphs are multiplicative.
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As well as being defined by recursive subdivision of triangles, the Apollonian networks have several other equivalent mathematical characterizations. They are the chordal maximal planar graphs, the chordal polyhedral graphs, and the planar 3-trees. They are the uniquely 4-colorable planar graphs, and the planar graphs with a unique Schnyder wood decomposition into three trees. They are the maximal planar graphs with treewidth three, a class of graphs that can be characterized by their forbidden minors or by their reducibility under Y-Δ transforms.
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All snarks are non-Hamiltonian, and many known snarks are hypohamiltonian: the removal of any single vertex leaves a Hamiltonian subgraph. A hypohamiltonian snark must be bicritical: the removal of any two vertices leaves a 3-edge-colorable subgraph. It has been shown that the number of snarks for a given even number of vertices is bounded below by an exponential function. (Being cubic graphs, all snarks must have an even number of vertices.) OEIS sequence contains the number of non-trivial snarks of 2n vertices for small values of n.
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Given an integer n, a hypergraph is called n-uniform if all its hyperedges contain exactly n vertices. An n-uniform hypergraph is called n-partite if its vertex set V can be partitioned into n subsets such that each hyperedge contains exactly one element from each subset. An alternative term is rainbow-colorable. To see that n-partiteness is stronger than exact-2-colorability, let H be a hypergraph on the vertices {1, 2, 3, 4} with the following hyperedges; > { {1,2,3} , {1,2,4} , {1,3,4} } This H is 3-uniform.
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Each of the seven color classes has one edge from the center to a polygon vertex, and three edges perpendicular to it. A complete graph with vertices is edge-colorable with colors when is an even number; this is a special case of Baranyai's theorem. provides the following geometric construction of a coloring in this case: place points at the vertices and center of a regular -sided polygon. For each color class, include one edge from the center to one of the polygon vertices, and all of the perpendicular edges connecting pairs of polygon vertices.
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The proof is computer assisted. Mathematician Gil Kalai and computer scientist Scott Aaronson posted discussion of de Grey's finding, with Aaronson reporting independent verifications of de Grey's result using SAT solvers. Kalai linked additional posts by Jordan Ellenberg and Noam Elkies, with Elkies and (separately) de Grey proposing a Polymath project to find non-4-colorable unit distance graphs with fewer vertices than the one in de Grey's construction. As of 2018, the smallest known graph with chromatic number 5 had 553 vertices , but in August 2019 Jaan Parts found a 510-vertex example.
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The crux of this test involved determining whether there is "more than a colorable difference" between the original infringing product and the redesign. EchoStar claimed significant changes to their DVR system in order to be compliant with the ruling. The key changes to the system were replacing the infringing event detection system with "start code detection was replaced by statistical estimation, a method which EchoStar characterized as an indexless system and a brute-force search method." The court determined that EchoStar Corp's software redesign still infringed TiVo Inc.
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A triangle-free penny graph with the property that all the pennies on the convex hull touch at least three other pennies Analogously, the degeneracy of every triangle-free penny graph is at most two. Every such graph contains a vertex with at most two neighbors, even though it is not always possible to find this vertex on the convex hull. Based on this one can prove that they require at most three colors, more easily than the proof of the more general Grötzsch's theorem that triangle-free planar graphs are 3-colorable.
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A 3-coloring of a graph G may be described by a graph homomorphism from G to a triangle K3. In the language of homomorphisms, Grötzsch's theorem states that every triangle-free planar graph has a homomorphism to K3. Naserasr showed that every triangle-free planar graph also has a homomorphism to the Clebsch graph, a 4-chromatic graph. By combining these two results, it may be shown that every triangle-free planar graph has a homomorphism to a triangle-free 3-colorable graph, the tensor product of K3 with the Clebsch graph.
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The hallmark of extraordinary constitutional jurisdiction is to keep various functionaries of State within the ambit of their authority. Once a High Court has assumed jurisdiction to adjudicate the matter before it, justiciability of the issue raised before it is beyond question. The Supreme Court of Pakistan has stated clearly that the use of words "in an unlawful manner" implies that the court may examine, if a statute has allowed such detention, whether it was a colorable exercise of the power of authority. Thus, the court can examine the malafides of the action taken.
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First, if planar regions separated by the graph are not triangulated, i.e. do not have exactly three edges in their boundaries, we can add edges without introducing new vertices in order to make every region triangular, including the unbounded outer region. If this triangulated graph is colorable using four colors or fewer, so is the original graph since the same coloring is valid if edges are removed. So it suffices to prove the four color theorem for triangulated graphs to prove it for all planar graphs, and without loss of generality we assume the graph is triangulated.
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The Court sometimes claims that the ability to declare constitutional rights is the most important power the federal judiciary wields. But many individual rights depend on administrative and statutory claims. Justice Antonin Scalia has argued that not "every constitutional claim is ipso facto more worthy, and every statutory claim less worthy, of judicial review." n202 Webster v. Doe, 486 U.S. 592 (1988) (Scalia, J., dissenting) (employee of Central Intelligence Agency sought review under the Administrative Procedure Act of his dismissal alleging constitutional and other violations; Court concluded that Congress did not intend to preclude judicial review of colorable constitutional challenges).
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But is not balanced. For example, if vertex 1 is removed, we get the restriction of H to {2,3,4}, which has the following hyperedges; > { {2,3} , {2,4} , {3,4} } It is not 2-colorable, since in any 2-coloring there are at least two vertices with the same color, and thus at least one of the hyperedges is monochromatic. Another way to see that H is not balanced is that it contains the odd-length cycle C = (2 - {1,2,3} - 3 - {1,3,4} - 4 - {1,2,4} - 2), and no edge of C contains all three vertices 2,3,4 of C. Balance does not imply bipartiteness.
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During his majority opinion holding for the petitioner, Justice Potter Stewart noted that "a colorable argument could be made" that Wilko did not reach the 1934 Act due to differences in the corresponding sections of the statutes. The non-waiver provision was narrower, the private right of action only implied, and the 1934 Act only provided for federal jurisdiction. Stewart found those concerns irrelevant to the case, since it concerned an international transaction under which arbitration might be a better solution.Scherk at 513–14, Stewart, J. The question had not been before the Court in Dean Witter Reynolds Inc. v.
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"At the outset", he wrote, "it is useful to review the manner by which the issue decided today has been kept alive inappropriately by this Court." He relegated his discussion of and response to Stewart's "colorable argument" to his footnotesShearson at 244n1, 245n2. since the majority had not, he claimed, relied on it and in any event "[i]t simply constituted a way of keeping the issue of the arbitrability of [1934 Act] claims alive for those opposed to the result in Wilko." Of greater relevance was Congress's refusal to address Wilko during the process of drafting and passing the 1975 amendments.
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Franchise Tax Board Test: The Court has also granted jurisdiction "where the vindication of a right under state law necessarily turned on some construction of federal law.". Smith Test (Quoting from the dissent): "The general rule is that where it appears from the bill or statement of the plaintiff that the right to relief depends upon the construction or application of the Constitution or laws of the United States, and that such federal claim is not merely colorable, and rests upon a reasonable foundation, the District Court has jurisdiction [478 U.S. 804, 820] under [the statute granting federal question jurisdiction].".
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Two greedy colorings of the same crown graph using different vertex orders. The right example generalises to 2-colorable graphs with vertices, where the greedy algorithm expends colors. In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but they do not in general use the minimum number of colors possible.
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A noncollinear placement of n points can also be interpreted as a graph drawing of the complete graph in such a way that, although edges cross, no edge passes through a vertex. Erdős' construction above can be generalized to show that every n-vertex k-colorable graph has such a drawing in a O(n) × O(k) grid . One can also consider graph drawings in the three-dimensional grid. Here the non-collinearity condition means that a vertex should not lie on a non- adjacent edge, but it is normal to work with the stronger requirement that no two edges cross (; ; ).
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United States, 417 U.S. 333, 346-47 (1974) ("There can be no room for doubt that such a circumstance 'inherently results in a complete miscarriage of justice' and 'present[s] exceptional circumstances' that justify collateral relief ...."); see also Satterfield v. Dist. Att'y of Phila., 872 F.3d 152, 164 (3d Cir. 2017) ("The fact that ... proceeding ended a decade ago should not preclude him from obtaining relief under Rule 60(b) if the court concludes that he has raised a colorable claim that he meets this threshold actual-innocence standard ...."); Herring v. United States, 424 F.3d 384, 386-87 (3d Cir.
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A different proof using Zorn's lemma was given by Lajos Pósa, and also in the 1951 Ph.D. thesis of Gabriel Andrew Dirac. If is an infinite graph in which every finite subgraph is -colorable, then by Zorn's lemma it is a subgraph of a maximal graph with the same property (one to which no more edges may be added without causing some finite subgraph to require more than colors). The binary relation of nonadjacency in is an equivalence relation, and its equivalence classes provide a -coloring of . However, this proof is more difficult to generalize than the compactness proof.
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Therefore, by the De Bruijn–Erdős theorem, P itself also has a w-colorable incomparability graph, and thus has the desired partition into chains . However, the theorem does not extend so simply to partially ordered sets in which the width, and not just the cardinality of the set, is infinite. In this case the size of the largest antichain and the minimum number of chains needed to cover the partial order may be very different from each other. In particular, for every infinite cardinal number κ there is an infinite partially ordered set of width ℵ0 whose partition into the fewest chains has κ chains .
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The inability of plaintiff > to recover an amount adequate to give the court jurisdiction does not show > his bad faith or oust the jurisdiction. Nor does the fact that the complaint > discloses the existence of a valid defense to the claim. But if, from the > face of the pleadings, it is apparent to a legal certainty that the > plaintiff cannot recover the amount claimed or if, from the proofs, the > court is satisfied to a like certainty that the plaintiff never was entitled > to recover that amount, and that his claim was therefore colorable for the > purpose of conferring jurisdiction, the suit will be dismissed.St. Paul > Mercury Indemnity Co. v.
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Potter Stewart, who wrote for the majority, based his holding on the international nature of the transaction. But he also briefly entertained a "colorable argument" that Wilko might not apply to the 1934 Act because of the differences in the corresponding sections, such as its limitation of the choice of forum to only federal courts as opposed to the state-court jurisdiction allowed under the 1933 Act.Scherk v. Alberto-Culver Co. (hereafter Scherk II), , at 513–514, Stewart, J. "Wilko was held by the Court of Appeals to control this case—and properly so," countered William O. Douglas, the only justice remaining from the Wilko Court, in his dissent.
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Upon the arrest of Ohio businessmen Paul Monea and Mickey Miller, its actual ownership was unclear. The Federal Bureau of Investigation seized the diamond following an undercover investigation of Monea and Miller that resulted in his conviction for money laundering and the diamond's forfeiture to the U.S. government in 2007. Amid Monea and Millers prosecution, 17 claimants came forward in the case to claim ownership, or an ownership interest, in the Golden Eye Diamond (Akron Federal Court Case No. 5:07CR0030). The Court ruled that three claimants had colorable ownership claims and would be allowed to proceed to civil trial: Charity Fellowship Church, Jerry DeLeo of Corona Clay Corp.
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Att'y of Phila., 872 F.3d 152, 164 (3d Cir. 2017) ("The fact that . . . proceeding ended a decade ago should not preclude him from obtaining relief under Rule 60(b) if the court concludes that he has raised a colorable claim that he meets this threshold actual- innocence standard ...."); see also Bousley v. United States, 523 U.S. 614, 622 (1998); United States v. Olano, 507 U.S. 725, 736 (1993); Davis v. United States, 417 U.S. 333, 346-47 (1974); Gonzalez-Cantu v. Sessions, 866 F.3d 302, 306 (5th Cir. 2017) (same) (collecting cases); Pacheco-Miranda v. Sessions, No. 14-70296 (9th Cir. Aug. 11, 2017) (same).
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Here, vertices of G are represented by straight line segments and edges of G are represented by intersection points. 240px 240px Scheinerman also conjectured that segments with only three directions would be sufficient to represent 3-colorable graphs, and conjectured that analogously every planar graph could be represented using four directions. If a graph is represented with segments having only k directions and no two segments belong to the same line, then the graph can be colored using k colors, one color for each direction. Therefore, if every planar graph can be represented in this way with only four directions, then the four color theorem follows.
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On the other hand, given a homomorphism G → H between undirected graphs, any orientation of H can be pulled back to an orientation of G so that has a homomorphism to . Therefore, a graph G is k-colorable (has a homomorphism to Kk) if and only if some orientation of G has a homomorphism to k. A folklore theorem states that for all k, a directed graph G has a homomorphism to k if and only if it admits no homomorphism from the directed path k+1. Here n is the directed graph with vertices 1, 2, …, n and edges from i to i + 1, for i = 1, 2, …, n − 1.
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The question before the court was whether Bucci's use of the PLANNED PARENTHOOD trademark in the website's domain name violated the Lanham Act's provisions. Trademark Infringement: Federal trademark law makes it a violation for a party to "use in commerce any reproduction, counterfeit, copy, or colorable imitation of a registered mark in connection with the sale, offering for sale, distribution, or advertising of any goods or services on or in connection with which such use is likely to cause confusion, or to cause mistake, or to deceive." . The court interpreted the "Use in Commerce" provision broadly to be co-extensive with Congress' commerce clause power under the Constitution.
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If the four-color conjecture were false, there would be at least one map with the smallest possible number of regions that requires five colors. The proof showed that such a minimal counterexample cannot exist, through the use of two technical concepts:; ; # An unavoidable set is a set of configurations such that every map that satisfies some necessary conditions for being a minimal non-4-colorable triangulation (such as having minimum degree 5) must have at least one configuration from this set. # A reducible configuration is an arrangement of countries that cannot occur in a minimal counterexample. If a map contains a reducible configuration, the map can be reduced to a smaller map.
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Att'y of Phila., 872 F.3d 152, 164 (3d Cir. 2017) ("The fact that ... proceeding ended a decade ago should not preclude him from obtaining relief under Rule 60(b) if the court concludes that he has raised a colorable claim that he meets this threshold actual-innocence standard ...."); see also Bousley v. United States, 523 U.S. 614, 622 (1998); United States v. Olano, 507 U.S. 725, 736 (1993) ("In our collateral review jurisprudence, the term 'miscarriage of justice' means that the defendant is actually innocent.... The court of appeals should no doubt correct a plain forfeited error that causes the conviction or sentencing of an actually innocent defendant....") (citations omitted); Davis v.
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The conjecture is studied in the more general context of graph homomorphisms, especially because of interesting relations to the category of graphs (with graphs as objects and homomorphisms as arrows). For any fixed graph K, one considers graphs G that admit a homomorphism to K, written G → K. These are also called K-colorable graphs. This generalizes the usual notion of graph coloring, since it follows from definitions that a k-coloring is the same as a Kk-coloring (a homomorphism into the complete graph on k vertices). A graph K is called multiplicative if for any graphs G, H, the fact that G × H → K holds implies that G → K or H → K holds.
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For a regular graph of degree that does not have a perfect matching, this lower bound can be used to show that at least colors are needed. In particular, this is true for a regular graph with an odd number of vertices (such as the odd complete graphs); for such graphs, by the handshaking lemma, must itself be even. However, the inequality does not fully explain the chromatic index of every regular graph, because there are regular graphs that do have perfect matchings but that are not k-edge-colorable. For instance, the Petersen graph is regular, with and with edges in its perfect matchings, but it does not have a 3-edge-coloring.
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Given a graph G and given a set L(v) of colors for each vertex v (called a list), a list coloring is a choice function that maps every vertex v to a color in the list L(v). As with graph coloring, a list coloring is generally assumed to be proper, meaning no two adjacent vertices receive the same color. A graph is k-choosable (or k-list-colorable) if it has a proper list coloring no matter how one assigns a list of k colors to each vertex. The choosability (or list colorability or list chromatic number) ch(G) of a graph G is the least number k such that G is k-choosable.
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For a graph G, let χ(G) denote the chromatic number and Δ(G) the maximum degree of G. The list coloring number ch(G) satisfies the following properties. # ch(G) ≥ χ(G). A k-list-colorable graph must in particular have a list coloring when every vertex is assigned the same list of k colors, which corresponds to a usual k-coloring. # ch(G) cannot be bounded in terms of chromatic number in general, that is, there is no function f such that ch(G) ≤ f(χ(G)) holds for every graph G. In particular, as the complete bipartite graph examples show, there exist graphs with χ(G) = 2 but with ch(G) arbitrarily large.
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In this article, unless stated otherwise, graphs are finite, undirected graphs with loops allowed, but multiple edges (parallel edges) disallowed. A graph homomorphismFor introductions, see (in order of increasing length): ; ; . f from a graph G = (V(G), E(G)) to a graph H = (V(H), E(H)), written : is a function from V(G) to V(H) that maps endpoints of each edge in G to endpoints of an edge in H. Formally, {u,v} ∈ E(G) implies {f(u),f(v)} ∈ E(H), for all pairs of vertices u, v in V(G). If there exists any homomorphism from G to H, then G is said to be homomorphic to H or H-colorable.
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Yoshio Shimamoto was a nuclear physicist who also did work in mathematics and computer science. While at Brookhaven National Laboratory (1954-1987), he designed the logic for the MERLIN digital computer in 1958, and served as chairman of the Applied Mathematics Department from 1964 to 1975. Shimamoto researched in combinatorial mathematics, the economics of outer continental shelf oil and gas lease sales (on behalf of the U.S. Geological Survey), the architecture of supercomputers, and the linking of computers for parallel processing. During the 1970s, he worked with Heinrich Heesch and Karl Durre on methods for a computer-aided proof of the four color theorem, using computer programs to apply Heesch's notion of "discharging" to eliminate 4-colorable cases.
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Yale Law School professor Bruce Ackerman wrote that the declaration of a national emergency to build a wall as Trump suggested would be unconstitutional and illegal.Bruce Ackerman, No, Trump Cannot Declare an 'Emergency' to Build His Wall , The New York Times (January 5, 2019). Other scholars, such as Elizabeth Goitein of the Brennan Center for Justice, believed that Trump could make a colorable argument that diverting military-construction appropriations for border-wall construction was legal, but that doing so would be an abuse of power. Law professor Ilya Somin stated that in the unlikely case that Trump succeeded in using the emergency powers in this way, it would set a dangerous precedent, which Republicans would come to regret next time the president was a Democrat.
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2017) ("The fact that ... proceeding ended a decade ago should not preclude him from obtaining relief under Rule 60(b) if the court concludes that he has raised a colorable claim that he meets this threshold actual- innocence standard...."); see also United States v. Olano, 507 U.S. 725, 736 (1993) ("In our collateral review jurisprudence, the term 'miscarriage of justice' means that the defendant is actually innocent.... The court of appeals should no doubt correct a plain forfeited error that causes the conviction or sentencing of an actually innocent defendant....") (citations omitted); Davis v. United States, 417 U.S. 333, 346-47 (1974) (regarding "miscarriage of justice" and "exceptional circumstances"); Gonzalez-Cantu v. Sessions, 866 F.3d 302, 306 (5th Cir.
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We record here definitions from . Let be the Coxeter complex associated to a group W generated by a set of reflections S. The vertices of are the elements of W, and the chambers of the complex are the cosets of S in W. The vertices of each chamber can be colored in a one-to-one manner by the elements of S so that no adjacent vertices of the complex receive the same color. This coloring, although essentially canonical, is not quite unique. The coloring of a given chamber is not uniquely determined by its realization as a coset of S. But once the coloring of a single chamber has been fixed, the rest of the Coxeter complex is uniquely colorable.
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Again, a double cover of the resulting graph may be extended in a straightforward way to a double cover of the original graph: every cycle of the split off graph corresponds either to a cycle of the original graph, or to a pair of cycles meeting at v. Thus, every minimal counterexample must be cubic. But if a cubic graph can have its edges 3-colored (say with the colors red, blue, and green), then the subgraph of red and blue edges, the subgraph of blue and green edges, and the subgraph of red and green edges each form a collection of disjoint cycles that together cover all edges of the graph twice. Therefore, every minimal counterexample must be a non-3-edge-colorable bridgeless cubic graph, that is, a snark.
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The defendants claimed two arguments of transformative qualities in The SAT: first that the subject matter of the work did not bar a finding of fair use, and second that the book was a critique of the show. Specifically, the defendants stylized The SAT as a work "decoding the obsession with and mystique that surrounds Seinfeld by critically restructuring Seinfeld's mystique into a system complete with varying levels of 'mastery' that relate the reader's control of the show's trivia to knowledge of and identification with their hero, Jerry Seinfeld."Castle Rock at 142. In response, Castle Rock retorted that "had defendants been half as creative in creating The SAT as were their lawyers in crafting these arguments about transformation, defendants might have a colorable fair use claim."Id.
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The existence of the Grötzsch graph demonstrates that the assumption of planarity is necessary in Grötzsch's theorem that every triangle-free planar graph is 3-colorable. used a modified version of the Grötzsch graph to disprove a conjecture of on the chromatic number of triangle-free graphs with high degree. Häggkvist's modification consists of replacing each of the five degree-four vertices of the Grötzsch graph by a set of three vertices, replacing each of the five degree-three vertices of the Grötzsch graph by a set of two vertices, and replacing the remaining degree-five vertex of the Grötzsch graph by a set of four vertices. Two vertices in this expanded graph are connected by an edge if they correspond to vertices connected by an edge in the Grötzsch graph.
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Antecedently, he was an austere vocal critic of indigenous practices that placed the wellbeing of native communities in peril and easy prey to the quackery of guileful practitioners—he worked towards getting those charlatanism practices extirpated. Moreover, he advocated for regulating native ethno-medicinal practices and outlawing those that were insanitary or insalubrious through erudition programs tailored to specific native communities’ socioculturalism. Congruently, he encouraged a scientific approach to traditional medicinal modalities, vis-à-vis, enacting of quality control criteria such as dosage guidelines in conjunction with promoting proven evidence-based time-tested and outcome-driven ethno-medicine. He presciently cognized that this could only be achieved through colorable scientifically modeled studies to authenticate the safety and efficacies of indigenous healing methods akin to the European or westernized medicine.
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The number of edges in an n-vertex linkless graph is at most 4n − 10: maximal apex graphs with n > 4 have exactly this many edges, and proved a matching upper bound on the more general class of K6-minor-free graphs. observed that Sachs' question about the chromatic number would be resolved by a proof of Hadwiger's conjecture that any k-chromatic graph has as a minor a k-vertex complete graph. The proof by of the case k = 6 of Hadwiger's conjecture is sufficient to settle Sachs' question: the linkless graphs can be colored with at most five colors, as any 6-chromatic graph contains a K6 minor and is not linkless, and there exist linkless graphs such as K5 that require five colors. The snark theorem implies that every cubic linklessly embeddable graph is 3-edge-colorable.
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He suggests that establishes a colorable Jewish —as well as Arab — claim to all of Palestine which tends to refute Professor Quigley's contention that there are no other claimants to that territory. Ash says there are segments of Israeli society that continue to view "Judea and Samaria" as areas promised to the Jews by the Balfour Declaration and says that the Geneva Convention is not applicable to Israel's presence in those territories. He cites Yehuda Blum's "Missing Reversioner" and Eugene Rostow's related claim that "The right of the Jewish people to settle in Palestine has never been terminated for the West Bank." He also notes that 'the terms of the Interim Agreement prohibit both Israel and the PA from “initiat[ing] or tak[ing] any step that will change the status of the West Bank and the Gaza Strip'.
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The record store had conceded that while the village had a legitimate governmental interest in curbing the sale and use of illegal drugs in its jurisdiction, no compelling state interest existed to justify abridgement of its First Amendment rights by the ordinance. "[T]he court is constrained to agree with defendants," wrote Leighton, since the Flipside have overcome the presumption that the ordinance was valid only if it had shown the absence of a rational basis. Since it had already conceded the village's interest in enforcing state drug laws, and Leighton had already found the ordinance did not infringe the record store's First Amendment rights, there was no constitutional violation. " Furthermore, there is no conceivable colorable claim of a fundamental constitutional right to sell items which facilitate and encourage the use of illegal drugs," Leighton added.
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Dilworth's theorem for infinite partially ordered sets states that a partially ordered set has finite width w if and only if it may be partitioned into w chains. For, suppose that an infinite partial order P has width w, meaning that there are at most a finite number w of elements in any antichain. For any subset S of P, a decomposition into w chains (if it exists) may be described as a coloring of the incomparability graph of S (a graph that has the elements of S as vertices, with an edge between every two incomparable elements) using w colors; every color class in a proper coloring of the incomparability graph must be a chain. By the assumption that P has width w, and by the finite version of Dilworth's theorem, every finite subset S of P has a w-colorable incomparability graph.
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Att'y of Phila., 872 F.3d 152, 164 (3d Cir. 2017) ("The fact that . . . proceeding ended a decade ago should not preclude him from obtaining relief under Rule 60(b) if the court concludes that he has raised a colorable claim that he meets this threshold actual-innocence standard ...."); see also United States v. Olano, 507 U.S. 725, 736 (1993) ("In our collateral review jurisprudence, the term 'miscarriage of justice' means that the defendant is actually innocent.... The court of appeals should no doubt correct a plain forfeited error that causes the conviction or sentencing of an actually innocent defendant....") (citations omitted); Davis v. United States, 417 U.S. 333, 346-47 (1974) (regarding "miscarriage of justice" and "exceptional circumstances"); Gonzalez-Cantu v. Sessions, 866 F.3d 302, 306 (5th Cir. 2017) (same); Pacheco- Miranda v. Sessions, No. 14-70296 (9th Cir. Aug. 11, 2017) (same).
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Justice William R. Day authored the majority opinion for Smith v. KC Title & Trust Co. The Court ruled that "even where a cause of action arises under state law, a federal court may have jurisdiction if it appears that the right to relief rests on the construction or application of a federal law." Justice Day ruled that the District Court had jurisdiction, stating, "the general rule is that where it appears from the bill or statement of the plaintiff that the right to relief depends upon the construction or application of the Constitution or laws of the United States, and that such federal claim is not merely colorable, and rests upon a reasonable foundation, the District Court has jurisdiction under this provision." So, since this case dealt with federal matters, the Supreme Court ruled that the District Court still has jurisdiction under certain circumstances.
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Should counsel's performance be judged by reference to a reasonable paid attorney or a reasonable appointed one? After all, Marshall pointed out, "a person of means, by selecting a lawyer and paying him enough to ensure he prepares thoroughly, usually can obtain better representation than that available to an indigent defendant, who must rely on appointed counsel, who, in turn, has limited time and resources to devote to a given case." Marshall also disputed that counsel's performance must be given especially wide latitude, since "much of the work involved in preparing for trial, applying for bail, conferring with one's client, making timely objections to significant, arguably erroneous rulings of the trial judge, and filing a notice of appeal if there are colorable grounds therefor could profitably be made the subject of uniform standards." Marshall also disputed that it should be made the defendant's burden to show prejudice from an allegedly incompetent attorney's performance.
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Aref's appeal, along with that of Hossain, was denied by the Second Circuit Court of Appeals on July 2, 2008. A petition for certiorari to the United States Supreme Court was denied on March 9, 2009. As reported on the front page of The New York Times on August 26, 2007, the Aref appeal could have been an important test case for the NSA warrantless wiretapping program, as it appeared to be the only criminal case where there was strong evidence that the program was used to target a defendantSpying Program May Be Tested By Terror Case, August 26, 2007 Both the ACLU and the NYCLU got involved in the case. However, in its opinion, the Second Circuit held that Aref had no right to know about this because it was classified. In an accompanying summary order, the court also sidestepped the issue by claiming that Aref hadn't shown a “colorable basis” for asserted he had been wiretapped under the NSA program.
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If existential quantification over vertices were also allowed, the resulting complexity class would be equal to NP (more precisely, the class of those properties of relational structures that are in NP), a fact known as Fagin's theorem. For example, SNP contains 3-Coloring (the problem of determining whether a given graph is 3-colorable), because it can be expressed by the following formula: : \exists S_1 \exists S_2 \exists S_3 \, \forall u \forall v \, \bigl( S_1(u) \vee S_2(u) \vee S_3(u) \bigr) \, \wedge \, \bigl( E(u,v)\,\implies\,( eg S_1(u) \vee eg S_1(v))\,\wedge\,\left( eg S_2(u) \vee eg S_2(v)\right)\,\wedge\,( eg S_3(u) \vee eg S_3(v)) \bigr) . Here E denotes the adjacency relation of the input graph, while the sets (unary relations) S_1,S_2,S_3 correspond to sets of vertices colored with one of the 3 colors. Similarly, SNP contains the k-SAT problem: the boolean satisfiability problem (SAT) where the formula is restricted to conjunctive normal form and to at most k literals per clause, where k is fixed.
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D. Pa. June 4, 2018); accord Satterfield v. Dist. Att'y of Phila., 872 F.3d 152, 164 (3d Cir. 2017) ("The fact that . . . proceeding ended a decade ago should not preclude him from obtaining relief under Rule 60(b) if the court concludes that he has raised a colorable claim that he meets this threshold actual-innocence standard ...."); see also United States v. Olano, 507 U.S. 725, 736 (1993) ("In our collateral review jurisprudence, the term 'miscarriage of justice' means that the defendant is actually innocent.... The court of appeals should no doubt correct a plain forfeited error that causes the conviction or sentencing of an actually innocent defendant....") (citations omitted); Davis v. United States, 417 U.S. 333, 346-47 (1974) (regarding "miscarriage of justice" and "exceptional circumstances"); Gonzalez-Cantu v. Sessions, 866 F.3d 302, 306 (5th Cir. 2017) (same); Pacheco- Miranda v. Sessions, No. 14-70296 (9th Cir. Aug. 11, 2017) (same). The review of the order does not require the alien (or the American) to remain in the United States.
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