Every Planar Map is Four Colorable

Appel, Kenneth E.; Haken, Wolfgang

doi:10.1090/conm/098

Cited by 351 publications

(402 citation statements)

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“…1). The extreme effectiveness of this coverage scheme is guaranteed by the well-known Four Colors Theorem, which establishes that "any map in a plane can be colored using four colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color" [7]. In particular, while three is the minimum number of different channels which can be used in any reuse scheme [5,6], four is a very common choice [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Satellite Multibeam Coverage of Earth: Innovative Solutions and Optimal Synthesis of Aperture Fields

Morabito¹,

Laganà²,

Donato³

2016

PIER

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Abstract-The problem of the synthesis of optimal continuous aperture sources to optimally realize the satellite multibeam coverage of Earth is stated and solved. The design approach relies on a far-field representation which exploits at best the degrees of freedom arising from the geometrical structure of the well-known four-colors coverage map. The overall synthesis is stated as a Convex Programming problem wherein the fast achievement of the (unique) globally optimal solution is guaranteed. The introduced tools allow stating the ultimate theoretical radiation performances achievable by any circular-aperture antenna of fixed size and, at the same time, can be exploited as a reference in the synthesis of isophoric direct-radiating arrays. Numerical examples concerning a mission scenario recently proposed by the European Space Agency are provided.

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Section: Introductionmentioning

confidence: 99%

Satellite Multibeam Coverage of Earth: Innovative Solutions and Optimal Synthesis of Aperture Fields

Morabito¹,

Laganà²,

Donato³

2016

PIER

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“…) Although not yet commonplace in theoretical results, computerassisted proofs are becoming more widespread, driven by the exponential growth of computing power which can be applied to previously and otherwise intractable problems. Nonetheless, as early as 1976, Appel and Haken proved the four color theorem [2,3] by using a computer program to check whether each of the thousands of possible candidates for the smallest-sized counter example to this theorem were actually four-colorable or not (simplified in [31]). Since then, important problems in various fields have been solved (fully or partially) with the assistance of computers: the discovery of Mersenne primes [39], the 17-point case of the happy ending problem [38], the NP-hardness of minimum-weight triangulation [26], a special case of Erdős' discrepancy conjecture [19], the ternary Goldbach conjecture [14], and Kepler's conjecture [13,24], among others.…”

Section: Introductionmentioning

confidence: 99%

Binary Pattern Tile Set Synthesis Is NP-hard

Kari

Kopecki

Meunier

et al. 2015

Automata, Languages, and Programming

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In the field of algorithmic self-assembly, a long-standing unproven conjecture has been that of the NP-hardness of binary pattern tile set synthesis (2-Pats). The k-Pats problem is that of designing a tile assembly system with the smallest number of tile types which will self-assemble an input pattern of k colors. Of both theoretical and practical significance, k-Pats has been studied in a series of papers which have shown k-Pats to be NP-hard for k = 60, k = 29, and then k = 11. In this paper, we close the fundamental conjecture that 2-Pats is NP-hard, concluding this line of study.While most of our proof relies on standard mathematical proof techniques, one crucial lemma makes use of a computer-assisted proof, which is a relatively novel but increasingly utilized paradigm for deriving proofs for complex mathematical problems. This tool is especially powerful for attacking combinatorial problems, as exemplified by the proof of the four color theorem by Appel and Haken (simplified later by Robertson, Sanders, Seymour, and Thomas) or the recent important advance on the Erdős discrepancy problem by Konev and Lisitsa using computer programs. We utilize a massively parallel algorithm and thus turn an otherwise intractable portion of our proof into a program which requires approximately a year of computation time, bringing the use of computer-assisted proofs to a new scale. We fully detail the algorithm employed by our code, and make the code freely available online.

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“…Even simple code becomes hard to manage if there is enough of it. The same holds for mathematical proof-the four-colour theorem famously had a proof too large for human referees to check [3]. The theorem was later formalised and proved in the interactive proof assistant Coq [4] by Gonthier [7,8], removing any doubt about its truth.…”

Section: Introductionmentioning

confidence: 99%

Challenges and Experiences in Managing Large-Scale Proofs

Bourke

Daum

Klein

et al. 2012

Lecture Notes in Computer Science

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Abstract. Large-scale verification projects pose particular challenges. Issues include proof exploration, efficiency of the edit-check cycle, and proof refactoring for documentation and maintainability. We draw on insights from two large-scale verification projects, L4.verified and Verisoft, that both used the Isabelle/HOL prover. We identify the main challenges in large-scale proofs, propose possible solutions, and discuss the Levity tool, which we developed to automatically move lemmas to appropriate theories, as an example of the kind of tool required by such proofs.

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Every Planar Map is Four Colorable

Cited by 351 publications

References 0 publications

Satellite Multibeam Coverage of Earth: Innovative Solutions and Optimal Synthesis of Aperture Fields

Satellite Multibeam Coverage of Earth: Innovative Solutions and Optimal Synthesis of Aperture Fields

Binary Pattern Tile Set Synthesis Is NP-hard

Challenges and Experiences in Managing Large-Scale Proofs

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