The minimum number of colours k required for which G is (k, d)-colourable is called the d-defective chromatic number, \chi_d (G).. For a graph class G, the defective chromatic number of G is minimum integer k such that for some integer d, every graph in G is (k,d)-colourable. For example, the defective chromatic number of the class of planar graphs equals 4, since every planar graph is (4,0)-colourable and for every integer d there is a planar graph that is not (3,d)-colourable.
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For every integer t there is an integer N such that every graph G with no K_t minor is (t-1,N)-colourable.
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Alternatively an interval edge coloring defined as: An edge-colouring of a graph G with colours 1. . . t is an 'interval t-colouring' if all colours are used, and the colours of edges incident to each vertex of G are distinct and form an interval of integers. A graph G is "interval colourable" if G has an interval t-colouring for some positive integer t. Let N be the set of all interval colourable graphs.
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This follows as if G is planar then every G_i (same as above) is outerplanar. Hence every G_i is (2,2)-colourable. Therefore, each vertex of V_i can be colored blue or red if i is even and green or yellow if i is odd, hence producing a (4,2)-coloring of G.
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The Court found the Act was not colourable. The evil that the law is addressing does not have to be approached directly, and in these circumstances it would not be practical. Even though the subject was not one that was commonly recognized as being criminal does not necessarily invalidate it.
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Triangle-free planar graphs are word- representable . A K4-free near-triangulation is 3-colourable if and only if it is word-representable M. Glen. Colourability and word-representability of near-triangulations, Pure and Applied Mathematics, to appear, 2019.; this result generalises studies in P. Akrobotu, S. Kitaev, and Z. Masárová.
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Major rejected the argument that the law was a "colourable" attempt to enact criminal law. The Act did not contain the penal consequences required for valid criminal law nor was there a criminal law purpose for the Act. The moral aspect to the law was only incidental to the regulatory scheme. Major rejected all of the Charter arguments.
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Therefore, at least four colors are needed to color this graph and the plane containing it. An alternative lower bound in the form of a ten-vertex four-chromatic unit distance graph, the Golomb graph, was discovered at around the same time by Solomon W. Golomb., p. 19. In 2018, computer scientist and biologist Aubrey de Grey found a 1581-vertex, non-4-colourable unit-distance graph.
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Therefore, the substance of the legislation must be articulated for the purpose of determining whether what it enacted, it could really do. The question of colourable legislation was fully discussed by the Supreme Court in K.C. Gajapati Narayan Deo v. Orissa, a decision which has been treated as settling the law on the subject. This ruling was confirmed in the Supreme Court decision of Sonapur Tea Co. v.
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Laskin CJ., in dissent, found that the law had no connection with Property and Civil Rights. Laskin identified the law as colourable. In form it concerned property but in substance it concerned the regulation of "taste". He noted how the province had already unsuccessfully attempted to prosecute the distributor of the film under the obscenity laws of the Criminal Code, and saw this as another attempt at the same goal.
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M/S R.M.D.C (Mysore) v. State of Mysore, (AIR 1962 SC 594) was a judgment of the Supreme Court of India, dealing with constitutional law, where the Court utilized the doctrine of colourable legislation. This is a landmark case on Centre-States relations in India, and was decided in favor of the erstwhile State of Mysore. This case dealt with issues of conflict between the States and Union regarding legislative competence.
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However, it later passed an ordinance amending certain provisions of the act to deem taxation powers upon themselves. This was called into question by the R.M.D.C. as being ultra vires the legislative competence of the State. It was also contended that the doctrine of colourable legislation was applicable, as the State of Mysore was attempting to control the conduct of prize competitions indirectly, by appropriating the powers of their taxation.
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Lindley LJ held that the transaction was wholly legitimate. He noted that Ooregum Gold Mining Co of India v Roper[1892] AC 125 decided shares cannot be issued at a discount, or below their nominal value, and continued. Smith LJ concurred, saying if the consideration is ‘not clearly colourable nor illusory, then, in my judgment, the adequacy of the consideration cannot be impeached by a liquidator unless the contract can also be impeached’.
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It held, obiter, that law might not encompass colourable legislation (that is, bills of attainder – legislation purporting to be of general application but in fact directed at securing the conviction of particular individuals), or legislation "of so absurd or arbitrary a nature that it could not possibly have been contemplated by our constitutional framers as being 'law' when they crafted the constitutional provisions protecting fundamental liberties".Yong Vui Kong (2010), p. 500, para. 16; see also pp.
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He found that the dominant feature of the Act was the "[protection] of the environment and human life and health from any and all harmful substances by regulating these substances." La Forest then considered whether the Act constituted "criminal law", which is a federal matter under section 91(27) of the Constitution Act, 1867. He observed that criminal law must contain high level of mens rea for true crimes. Laws can be disguised (known as "colourable law") as criminal in order to intrude on provincial authority.
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In addressing the municipality's argument that the residency requirement was merely a private employment contract and not a governmental function, La Forest J. found that once a body is labeled governmental, that body cannot use colourable devices or organize activities to avoid Charter responsibility. La Forest considered the validity of the law under section 7 of the Canadian Charter. He identified section 7 as protecting personal autonomy which includes the choice of selecting one's home. At no time did Godbout waive that right, even in signing the employment contract that contained the residency restriction.
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Finally, it was argued that the amendments were colourable legislation as the tax was imposed on the prize competitions with the objective of restoring their control to the State. They submitted that the state legislature were imposing penalties upon prize competitions by way of tax, and thus implementing control over such competitions. They further submitted that such taxation was a method of control and regulation, and all matters ancillary to the issue of control and regulation had been surrendered to the Union (via the doctrine of ancillary powers).
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A replica of the brown bag and the colourable line drawing are included in this edition. The reissue was met with generally positive reviews. At Metacritic, which assigns a normalized rating out of 100 to reviews from mainstream publications, the album received an average score of 73, based on eight reviews. Q magazine said "it's aged remarkably well and 'All My Love' is breathtakingly beautiful", while Tim Batcup from Classic Rock observed in the bonus material "a scruffier, rambunctious 'Hot Dog' and a sparser 'In the Evening', the drone intro truncated and Jones's synths high in the mix".
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The Court stated that the Act was ultra vires the province and so struck it down. In performing pith and substance analysis on the legislation by looking at the Act's purpose and effect, the Court found that the Act was colourable. That is, the form of the Act appears to address a valid matter but in substance actually addresses a matter outside its authority. The "Pith and Substance" of the Act, its dominant feature or purpose, was to interfere with the right of Hydro-Québec granted by the agreement with Churchill Falls Corp to receive power from across the provincial border.
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The effects can also reveal whether a law is "colourable", i.e. does the law in form appear to address something within the legislature's jurisdiction, but in substance deal with a matter outside that jurisdiction. para. 17 McLachlin says that the court must look at the wording of the Act as well as the circumstances that it was enacted. On the facts of the case the purpose of the regulation was :to regulate the seal fishery by eliminating the commercial hunting of whitecoats and bluebacks through a prohibition on the sale, while at the same time allowing for a limited harvesting of these animals for non-commercial purposes.
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This would interfere with Preston's preaching at Lincoln's Inn. His ingenuity found out evasions to which the fellows consented; the statutes condoned absence in case of 'violent detention ' and of 'college business'; a 'moral violence ' was held to satisfy the former condition, and a suit at law about a college living, which lasted some years, formed a colourable pretext for alleging college business. But Preston was inflexible on the point of vacating fellowships. According to Ball, he had been selected by Buckingham to accompany Arthur Chichester, 1st Baron Chichester, on a projected embassy to Germany, and was, on this occasion, was made a doctor of divinity by royal mandate.
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This > was simply a harassing tactic, for books were being conveyed to Elizabeth, > some of which Bedingfield suspected of being seditious, and when Parry sent > him two harmless ones he was forced to return them for want of explicit > instructions. Bedingfield complained that he was helpless, as ‘daily and > hourly the said Parry may have and give intelligence’, and once again the > cofferer's position was referred to the Council. Early in Jul Parry was at > the Bull inn, ‘a marvellous colourable place to practise in’, receiving > every day as many as 40 men in his own livery, besides Elizabeth's own > servants. At length the Council forbade such large meetings and, from > Bedingfield's subsequent silence on the point, it seems that the order was > obeyed.
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SCC, par. 37 They held that the fact that section 29 might hinder Quebec's efforts to create their own registry was insufficient to make out colourable legislation.SCC, par. 40 They clarified the pith and substance analysis by specifying that a legislative provision repealing a criminal offence would fall within the scope of the criminal law power, even though it is not strictly a law applying a prohibition and penalty for a public purpose.SCC, par. 33 Classifying the provision itself, Cromwell and Karakatsanis JJ found that it also fell within the federal criminal law power, since "The power to repeal a criminal law provision must logically be wide enough to give Parliament jurisdiction to destroy the data collected for the purpose of a criminal law provision".
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The Supreme Court held that the Central legislation, passed in accordance with article 252(1) of the Indian Constitution, governed entry 34 of List II of the Seventh Schedule, while the amendments made by the Mysore Legislation were done under entry 62 of the same list. These entries covered separate powers, and by passing the resolution under article 252, the States did not surrender their powers of taxation under entry 62. It could not be held that article 252(2) of the Indian Constitution was violated by such an amendment of the Mysore Act, nor could it be held that the amendment amounted to colourable legislation. as the amendment did not amount to indirect control and regulation of prize competitions.
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It was in this situation that the Privy Council evolved the doctrine that, for deciding whether an impugned legislation was intra vires, regard must be had to its pith and substance. Thus, if a statute is found in substance to relate to a topic within the competence of the legislature, it should be held to be intra vires even though it might incidentally trench on topics not within its legislative competence. The extent of the encroachment on matters beyond its competence may be an element in determining whether the legislation is colourable: whether in the guise of making a law on a matter within its competence, the legislature is, in truth, making a law on a subject beyond its competence. However, where that is not the position, the fact of encroachment does not affect the vires of the law even as regards the area of encroachment.
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The scope of Neilson's patent was questioned in this and other cases. What constituted a receptacle or reservoir? Some accused infringers used a network of pipes, which they claimed worked better than Neilson’s box-like receptacle, because of a greater surface area, which facilitated better heat transfer.See Web. Pat. Cases at 304 (defendant’s argument at trial, describing defendant’s apparatus). Baron Alderson Baron Alderson said that Neilson’s patent covered use of an “intermediate reservoir between the blast furnace and the bellows.” Therefore, “surely anybody else may apply the same principle, provided he does not do it by a reservoir intermediately between the blast furnace and the bellows.” It then became a fact question as to what constituted a “reservoir”. It was for the jury to decide, Baron Alderson said, “whether or not a long spiral pipe is a reservoir,” for “if it be not a reservoir, or a colourable imitation of a reservoir, it is no infringement.”See Neilson, Web. Pat.
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The President Robin Cooke, Baron Cooke of Thorndon, as he became (1995) was known to hold a personal copy of McHugh's dissertation and in conversation with him as well as having shown judicial mettle in the Maori Council cases of the time. Though based upon the claim to present unextinguished rights (rather than those lost by colourable Crown conduct in the past) this was the country's first multimillion-dollar settlement of Maori claims in the modern era, and a foretaste of the claims settlement processes and asset transfers that followed and as contemplated by the Maori Council cases. McHugh is associated with the view that common law aboriginal title was primarily a legal argument devised (in western Canada initially) during the early 1970s and following decade to deal with the inaction of the political branches (their incapacity to legislate comprehensive land rights/claims regimes). McHugh was amongst a small group of lawyer/scholars during the 1980s (Brian Slattery, Kent McNeil, Henry Reynolds, Barbara Hocking, Richard Bartlett, Tom Berger) to package aboriginal title in a manner that became palatable to courts.
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