An Exploration of Late Twentieth and Twenty-First Century Clarinet Repertoire
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A Brief History of Edge-Colorings — with Personal Reminiscences
Discrete Mathematics Letters Discrete Math. Lett. 6 (2021) 38–46 www.dmlett.com DOI: 10.47443/dml.2021.s105 Review Article A brief history of edge-colorings – with personal reminiscences∗ Bjarne Toft1;y, Robin Wilson2;3 1Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark 2Department of Mathematics and Statistics, Open University, Walton Hall, Milton Keynes, UK 3Department of Mathematics, London School of Economics and Political Science, London, UK (Received: 9 June 2020. Accepted: 27 June 2020. Published online: 11 March 2021.) c 2021 the authors. This is an open access article under the CC BY (International 4.0) license (www.creativecommons.org/licenses/by/4.0/). Abstract In this article we survey some important milestones in the history of edge-colorings of graphs, from the earliest contributions of Peter Guthrie Tait and Denes´ Konig¨ to very recent work. Keywords: edge-coloring; graph theory history; Frank Harary. 2020 Mathematics Subject Classification: 01A60, 05-03, 05C15. 1. Introduction We begin with some basic remarks. If G is a graph, then its chromatic index or edge-chromatic number χ0(G) is the smallest number of colors needed to color its edges so that adjacent edges (those with a vertex in common) are colored differently; for 0 0 0 example, if G is an even cycle then χ (G) = 2, and if G is an odd cycle then χ (G) = 3. For complete graphs, χ (Kn) = n−1 if 0 0 n is even and χ (Kn) = n if n is odd, and for complete bipartite graphs, χ (Kr;s) = max(r; s). -
Back Cover Image
559682 rr Reich EU.qxp_559682 rr Reich EU 18/09/2020 08:44 Page 1 CMYK N 2 A 8 Steve X 6 Pl aying O 9 Time : S 5 5 R(bE. I19C36)H . 73:24 1 B o U A 8 f o l n l t o a Music for Two or More Pianos 10:57 h r (1964)* k i u 2 i g MERICAN LASSICS s l A C t e h h t c t o s Eight Lines for ensemble (1979/1983) 17:12 o n r 3 o m i i z n t Steve Reich is universally acknowledged e e p t S s d Vermont Counterpoint a h c i i T p s as one of the foremost exponents of n t u 9:33 for flutes and tape (1982) d s E E b o i n s l minimalism, arguably the most i u V e c c g n f New York Counterpoint l i p p d i s significant stylistic trend in late 20th- E r e h o r r L 4 12:07 e for clarinets and tape (1985) h f • o R c century music. This chronological i K = ca. 184 4:56 b r 5 o y m o i E r t t d e m = ca. 92 2:40 i a 6 survey shows how Reich’s innate i d I n n m . = ca. 184 3:31 C C c g e ൿ e curiosity has taken his work far beyond , n • , H a t & b a r s t such musical boundaries. -
On the Cycle Double Cover Problem
On The Cycle Double Cover Problem Ali Ghassâb1 Dedicated to Prof. E.S. Mahmoodian Abstract In this paper, for each graph , a free edge set is defined. To study the existence of cycle double cover, the naïve cycle double cover of have been defined and studied. In the main theorem, the paper, based on the Kuratowski minor properties, presents a condition to guarantee the existence of a naïve cycle double cover for couple . As a result, the cycle double cover conjecture has been concluded. Moreover, Goddyn’s conjecture - asserting if is a cycle in bridgeless graph , there is a cycle double cover of containing - will have been proved. 1 Ph.D. student at Sharif University of Technology e-mail: [email protected] Faculty of Math, Sharif University of Technology, Tehran, Iran 1 Cycle Double Cover: History, Trends, Advantages A cycle double cover of a graph is a collection of its cycles covering each edge of the graph exactly twice. G. Szekeres in 1973 and, independently, P. Seymour in 1979 conjectured: Conjecture (cycle double cover). Every bridgeless graph has a cycle double cover. Yielded next data are just a glimpse review of the history, trend, and advantages of the research. There are three extremely helpful references: F. Jaeger’s survey article as the oldest one, and M. Chan’s survey article as the newest one. Moreover, C.Q. Zhang’s book as a complete reference illustrating the relative problems and rather new researches on the conjecture. A number of attacks, to prove the conjecture, have been happened. Some of them have built new approaches and trends to study. -
ZGMTH - the Order of Things
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Bangor University Research Portal The Order of Things. Analysis and Sketch Study in Two Works by Steve ANGOR UNIVERSITY Reich Bakker, Twila; ap Sion, Pwyll Zeitschrift der Gesellschaft für Musiktheorie DOI: 10.31751/1003 PRIFYSGOL BANGOR / B Published: 01/06/2019 Publisher's PDF, also known as Version of record Cyswllt i'r cyhoeddiad / Link to publication Dyfyniad o'r fersiwn a gyhoeddwyd / Citation for published version (APA): Bakker, T., & ap Sion, P. (2019). The Order of Things. Analysis and Sketch Study in Two Works by Steve Reich. Zeitschrift der Gesellschaft für Musiktheorie, 16(1), 99-122. https://doi.org/10.31751/1003 Hawliau Cyffredinol / General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. 09. Oct. 2020 ZGMTH - The Order of Things https://www.gmth.de/zeitschrift/artikel/1003.aspx Inhalt (/zeitschrift/ausgabe-16-1-2019/inhalt.aspx) Impressum (/zeitschrift/ausgabe-16-1-2019/impressum.aspx) Autorinnen und Autoren (/zeitschrift/ausgabe-16-1-2019/autoren.aspx) Home (/home.aspx) Bakker, Twila / ap Siôn, Pwyll (2019): The Order of Things. -
Graph Algorithms Graph Algorithms a Brief Introduction 3 References 高晓沨 ((( Xiaofeng Gao )))
目录 1 Graph and Its Applications 2 Introduction to Graph Algorithms Graph Algorithms A Brief Introduction 3 References 高晓沨 ((( Xiaofeng Gao ))) Department of Computer Science Shanghai Jiao Tong Univ. 2015/5/7 Algorithm--Xiaofeng Gao 2 Konigsberg Once upon a time there was a city called Konigsberg in Prussia The capital of East Prussia until 1945 GRAPH AND ITS APPLICATIONS Definitions and Applications Centre of learning for centuries, being home to Goldbach, Hilbert, Kant … 2015/5/7 Algorithm--Xiaofeng Gao 3 2015/5/7 Algorithm--Xiaofeng Gao 4 Position of Konigsberg Seven Bridges Pregel river is passing through Konigsberg It separated the city into two mainland area and two islands. There are seven bridges connecting each area. 2015/5/7 Algorithm--Xiaofeng Gao 5 2015/5/7 Algorithm--Xiaofeng Gao 6 Seven Bridge Problem Euler’s Solution A Tour Question: Leonhard Euler Solved this Can we wander around the city, crossing problem in 1736 each bridge once and only once? Published the paper “The Seven Bridges of Konigsbery” Is there a solution? The first negative solution The beginning of Graph Theory 2015/5/7 Algorithm--Xiaofeng Gao 7 2015/5/7 Algorithm--Xiaofeng Gao 8 Representing a Graph More Examples Undirected Graph: Train Maps G=(V, E) V: vertex E: edges Directed Graph: G=(V, A) V: vertex A: arcs 2015/5/7 Algorithm--Xiaofeng Gao 9 2015/5/7 Algorithm--Xiaofeng Gao 10 More Examples (2) More Examples (3) Chemical Models 2015/5/7 Algorithm--Xiaofeng Gao 11 2015/5/7 Algorithm--Xiaofeng Gao 12 More Examples (4) More Examples (5) Family/Genealogy Tree Airline Traffic 2015/5/7 Algorithm--Xiaofeng Gao 13 2015/5/7 Algorithm--Xiaofeng Gao 14 Icosian Game Icosian Game In 1859, Sir William Rowan Hamilton Examples developed the Icosian Game. -
Snarks and Flow-Critical Graphs 1
Snarks and Flow-Critical Graphs 1 CˆandidaNunes da Silva a Lissa Pesci a Cl´audioL. Lucchesi b a DComp – CCTS – ufscar – Sorocaba, sp, Brazil b Faculty of Computing – facom-ufms – Campo Grande, ms, Brazil Abstract It is well-known that a 2-edge-connected cubic graph has a 3-edge-colouring if and only if it has a 4-flow. Snarks are usually regarded to be, in some sense, the minimal cubic graphs without a 3-edge-colouring. We defined the notion of 4-flow-critical graphs as an alternative concept towards minimal graphs. It turns out that every snark has a 4-flow-critical snark as a minor. We verify, surprisingly, that less than 5% of the snarks with up to 28 vertices are 4-flow-critical. On the other hand, there are infinitely many 4-flow-critical snarks, as every flower-snark is 4-flow-critical. These observations give some insight into a new research approach regarding Tutte’s Flow Conjectures. Keywords: nowhere-zero k-flows, Tutte’s Flow Conjectures, 3-edge-colouring, flow-critical graphs. 1 Nowhere-Zero Flows Let k > 1 be an integer, let G be a graph, let D be an orientation of G and let ϕ be a weight function that associates to each edge of G a positive integer in the set {1, 2, . , k − 1}. The pair (D, ϕ) is a (nowhere-zero) k-flow of G 1 Support by fapesp, capes and cnpq if every vertex v of G is balanced, i. e., the sum of the weights of all edges leaving v equals the sum of the weights of all edges entering v. -
A Nonesuch Retrospective
STEVE REICH PHASES A N O nes U ch R etr O spective D isc One D isc T W O D isc T hree MUSIC FOR 18 MUSICIANS (1976) 67:42 DIFFERENT TRAINS (1988) 26:51 YOU ARE (VARIATIONS) (2004) 27:00 1. Pulses 5:26 1. America—Before the war 8:59 1. You are wherever your thoughts are 13:14 2. Section I 3:58 2. Europe—During the war 7:31 2. Shiviti Hashem L’negdi (I place the Eternal before me) 4:15 3. Section II 5:13 3. After the war 10:21 3. Explanations come to an end somewhere 5:24 4. Section IIIA 3:55 4. Ehmor m’aht, v’ahsay harbay (Say little and do much) 4:04 5. Section IIIB 3:46 Kronos Quartet 6. Section IV 6:37 David Harrington, violin Los Angeles Master Chorale 7. Section V 6:49 John Sherba, violin Grant Gershon, conductor 8. Section VI 4:54 Hank Dutt, viola Phoebe Alexander, Tania Batson, Claire Fedoruk, Rachelle Fox, 9. Section VII 4:19 Joan Jeanrenaud, cello Marie Hodgson, Emily Lin, sopranos 10. Section VIII 3:35 Sarona Farrell, Amy Fogerson, Alice Murray, Nancy Sulahian, 11. Section IX 5:24 TEHILLIM (1981) 30:29 Kim Switzer, Tracy Van Fleet, altos 12. Section X 1:51 4. Part I: Fast 11:45 Pablo Corá, Shawn Kirchner, Joseph Golightly, Sean McDermott, 13. Section XI 5:44 5. Part II: Fast 5:54 Fletcher Sheridan, Kevin St. Clair, tenors 14. Pulses 6:11 6. Part III: Slow 6:19 Geri Ratella, Sara Weisz, flutes 7. -
Signumclassics
143booklet 23/9/08 13:15 Page 1 ALSO AVAILABLE on signumclassics Different Trains Time for Marimba Steve Reich Daniella Ganeva The Smith Quartet SIGCD057 SIGCD064 Signum Classics are proud to release the Smith Quartet’s debut Time for Marimba explores the 20-th century Japanese repertoire for disc on Signum Records - Different Trains. The disc contains three marimba including music by five pre-eminent post-war composers. of Steve Reich’s most inspiring works: Triple Quartet for three string quartets, Reich’s personal dedication to the late Yehudi Menuhin, Duet, and the haunting Different Trains for string quartet and electronic tape. Available through most record stores and at www.signumrecords.com For more information call +44 (0) 20 8997 4000 143booklet 23/9/08 13:15 Page 3 Electric Counterpoint Electric Counterpoint Steve Reich 1. I Fast [6.51] 2. II Slow [3.22] 3. III Fast [4.33] 4. Tour de France Kraftwerk [5.25] 5. Radioactivity Kraftwerk [5.57] 6. Pocket Calculator Kraftwerk [6.44] 7. Carbon Copy Joby Burgess & Matthew Fairclough [5.14] 8. Temazcal Javier Alvarez [8.05] 9. Audiotectonics III Matthew Fairclough [3.47] Total Timings [49.55] Video: Temazcal [8.09] Powerplant Joby Burgess percussion Matthew Fairclough sound design Kathy Hinde visual artist The Elysian Quartet www.signumrecords.com 143booklet 23/9/08 13:15 Page 5 Electric Counterpoint Tour de France improvising with flute, electric guitar, cello and Carbon Copy Steve Reich (1936-) Ralf Hutter (1946-), Florian Schneider (1947-) vibraphone as the then Organisation, they Joby Burgess (1976-) and Matthew Fairclough I Fast II Slow III Fast and Karl Bartos (1952-) explored the sounds of the modern industrial (1970-) world, setting up their own studio, Kling Klang, Joby Burgess xylosynth Radioactivity which remains at a secret location in Dusseldorf. -
Generation of Cubic Graphs and Snarks with Large Girth
Generation of cubic graphs and snarks with large girth Gunnar Brinkmanna, Jan Goedgebeura,1 aDepartment of Applied Mathematics, Computer Science & Statistics Ghent University Krijgslaan 281-S9, 9000 Ghent, Belgium Abstract We describe two new algorithms for the generation of all non-isomorphic cubic graphs with girth at least k ≥ 5 which are very efficient for 5 ≤ k ≤ 7 and show how these algorithms can be efficiently restricted to generate snarks with girth at least k. Our implementation of these algorithms is more than 30, respectively 40 times faster than the previously fastest generator for cubic graphs with girth at least 6 and 7, respec- tively. Using these generators we have also generated all non-isomorphic snarks with girth at least 6 up to 38 vertices and show that there are no snarks with girth at least 7 up to 42 vertices. We present and analyse the new list of snarks with girth 6. Keywords: cubic graph, snark, girth, chromatic index, exhaustive generation 1. Introduction A cubic (or 3-regular) graph is a graph where every vertex has degree 3. Cubic graphs have interesting applications in chemistry as they can be used to represent molecules (where the vertices are e.g. carbon atoms such as in fullerenes [20]). Cubic graphs are also especially interesting in mathematics since for many open problems in graph theory it has been proven that cubic graphs are the smallest possible potential counterexamples, i.e. that if the conjecture is false, the smallest counterexample must be a cubic graph. For examples, see [5]. For most problems the possible counterexamples can be further restricted to the sub- class of cubic graphs which are not 3-edge-colourable. -
Introduction to the DFT and Music for Pieces of Wood
Cyclic Rhythm and Rhythmic Qualities: Introduction to the DFT and an Application to Steve Reich’s Phase Music Jason Yust, Spring 2018 Much of Steve Reich’s music is based on repetition (often ad lib) of 1–2 measure rhythmic patterns. This is particularly true of his “phase” music from Piano Phase and Violin Phase in 1967 to Octet [Eight Lines] in 1979, and also much of the music of the 1980s, such as New York Counterpoint and Electric Counterpoint. The repetition of a consistent unit, one or two measures of 4/4 or 12/8, sometimes 10/8, with a basic pulse of and eighth-note or sixteenth, creates the framework of a rhythmic cycle. Rhythmic cycles invite an analogy with pitch classes, which are also cyclic. Hence, we can think of cyclic rhythms as “beat-class sets.” (This idea was first suggested by Milton Babbitt and built upon by Jeff Pressing, Richard Cohn, and John Roeder.1) Here, I will propose a method of analyzing cyclic rhythms using the DFT (discrete Fourier transform). This is, mathematically, the same method that is used for determining what frequencies are present in an audio signal. The differences when using this method to analyze beat-class sets are (1) Order of magnitude: the frequencies are much slower, so that they correspond to rhythmic periodicities rather than pitches, and (2) Simplifying assumptions: Rhythmic cycles include a relatively small number of eighth or sixteenth-notes (typically 6, 8, 10, 12, 16, or 24), and they are assumed to be perfectly periodic, meaning they could repeat indefinitely. -
Beatrice Ruini
LE MATEMATICHE Vol. LXV (2010) – Fasc. I, pp. 3–21 doi: 10.4418/2010.65.1.1 SOME INFINITE CLASSES OF ASYMMETRIC NEARLY HAMILTONIAN SNARKS CARLA FIORI - BEATRICE RUINI We determine the full automorphism group of each member of three infinite families of connected cubic graphs which are snarks. A graph is said to be nearly hamiltonian if it has a cycle which contains all vertices but one. We prove, in particular, that for every possible order n ≥ 28 there exists a nearly hamiltonian snark of order n with trivial automorphism group. 1. Introduction Snarks are non-trivial connected cubic graphs which do not admit a 3-edge- coloring (a precise definition will be given below). The term snark owes its origin to Lewis Carroll’s famouse nonsense poem “The Hunting of the Snark”. It was introduced as a graph theoretical term by Gardner in [13] when snarks were thought to be very rare and unusual “creatures”. Tait initiated the study of snarks in 1880 when he proved that the Four Color Theorem is equivalent to the statement that no snark is planar. Asymmetric graphs are graphs pos- sessing a single graph automorphism -the identity- and for that reason they are also called identity graphs. Twenty-seven examples of asymmetric graphs are illustrated in [27]. Two of them are the snarks Sn8 and Sn9 of order 20 listed in [21] p. 276. Asymmetric graphs have been the subject of many studies, see, Entrato in redazione: 4 febbraio 2009 AMS 2000 Subject Classification: 05C15, 20B25. Keywords: full automorphism group, snark, nearly hamiltonian snark. -
SNARKS Generation, Coverings and Colourings
SNARKS Generation, coverings and colourings jonas hägglund Doctoral thesis Department of mathematics and mathematical statistics Umeå University April 2012 Doctoral dissertation Department of mathematics and mathematical statistics Umeå University SE-901 87 Umeå Sweden Copyright © 2012 Jonas Hägglund Doctoral thesis No. 53 ISBN: 978-91-7459-399-0 ISSN: 1102-8300 Printed by Print & Media Umeå 2012 To my family. ABSTRACT For a number of unsolved problems in graph theory such as the cycle double cover conjecture, Fulkerson’s conjecture and Tutte’s 5-flow conjecture it is sufficient to prove them for a family of graphs called snarks. Named after the mysterious creature in Lewis Carroll’s poem, a snark is a cyclically 4-edge connected 3-regular graph of girth at least 5 which cannot be properly edge coloured using three colours. Snarks and problems for which an edge minimal counterexample must be a snark are the central topics of this thesis. The first part of this thesis is intended as a short introduction to the area. The second part is an introduction to the appended papers and the third part consists of the four papers presented in a chronological order. In Paper I we study the strong cycle double cover conjecture and stable cycles for small snarks. We prove that if a bridgeless cubic graph G has a cycle of length at least jV(G)j - 9 then it also has a cycle double cover. Furthermore we show that there exist cyclically 5-edge connected snarks with stable cycles and that there exists an infinite family of snarks with stable cycles of length 24.