Brian Alspach and His Work
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Saad I. El-Zanati
SAAD I. EL-ZANATI Addresses: Office Home Department of Mathematics 2159 County Road 355N Illinois State University Carlock, IL 61725 Normal, IL 61790-4520 (309)438{5765 (309)376{3776 E-mail Address: [email protected] Web Site: http://math.illinoisstate.edu/saad/ Education: Ph.D., Mathematics, Auburn University (1991). M.S., Mathematics, Auburn University (1987). B.S., Computer Engineering, Auburn University (1983). Ph.D. Dissertation: Graph Designs, Auburn University (1991). Advisor: Christopher A. Rodger. Experience: • Professor, Department of Mathematics, Illinois State University, 2000{present. • Associate Professor, Department of Mathematics, Illinois State University, 1996{2000. • Assistant Professor, Department of Mathematics, Illinois State University, 1991{1996. • Instructor, Department of Mathematics, Tuskegee University, 1987{1988. • Graduate Teaching Assistant, Department of Mathematics, Auburn University, 1984{ 1991. Courses Taught at Illinois State University: (*indicates topics I developed) Calcu- lus I, II, & III, Elementary Linear Algebra, Introduction to Real Analysis, Introduction to Discrete Mathematics, Discrete Mathematics, Honors Undergraduate Research I & II, History of Math to 1600, Advanced Linear Algebra, Advanced Calculus, Topics in Discrete Mathematics: Coding Theory*, Topics in Discrete Mathematics: Design Theory*, Advanced Topics in Discrete Mathematics: Design Theory*, Graph Theory, Independent Study, Topics in Mathematics for Secondary School Teachers*, Introduction to Undergraduate Research in Mathematics*, Undergraduate Research in Mathematics II*, Summer Research in Math- ematics*, Mathematics for Secondary Teachers, Teaching Methods, Foundations of Inquiry. 1 Publications: (*indicates undergraduate co-author; **indicates graduate co-author) 74. On decompositions of complete multipartite graphs into the union of two even cycles, submitted. (with J. Buchanan*, R. C. Bunge**, E. Pelttari*, G. Rasmuson**, A. Su, and S. -
On Bipartite 2-Factorisations of K N − I and the Oberwolfach Problem
On bipartite 2-factorisations of Kn − I and the Oberwolfach problem Darryn Bryant ∗ Peter Danziger † The University of Queensland Department of Mathematics Department of Mathematics Ryerson University Qld 4072 Toronto, Ontario, Australia Canada M5B 2K3 Abstract It is shown that if F1,F2,...,Ft are bipartite 2-regular graphs of order n and n−2 α1, α2, . , αt are non-negative integers such that α1+α2+···+αt = 2 , α1 ≥ 3 is odd, and αi is even for i = 2, 3, . , t, then there exists a 2-factorisation of Kn −I in which there are exactly αi 2-factors isomorphic to Fi for i = 1, 2, . , t. This result completes the solution of the Oberwolfach problem for bipartite 2- factors. 1 Introduction The Oberwolfach problem was posed by Ringel in the 1960s and is first mentioned in [19]. It relates to specification of tournaments and specifically to balanced seating arrangments at round tables. In this article we will provide a complete solution to the Oberwolfach problem in the case where there are an even number of seats at each table. Let n ≥ 3 and let F be a 2-regular graph of order n. When n is odd, the Oberwol- fach problem OP(F ) asks for a 2-factorisation of the complete graph Kn on n vertices ∗Supported by the Australian Research Council †Supported by NSERC in which each 2-factor is isomorphic to F . When n is even, the Oberwolfach problem OP(F ) asks for a 2-factorisation of Kn − I, the complete graph on n vertices with the edges of a 1-factor removed, in which each 2-factor is isomorphic to F . -
The Oberwolfach Problem
Merging Combinatorial Design and Optimization: the Oberwolfach Problem Fabio Salassaa, Gabriele Dragottoc,a,∗, Tommaso Traettad, Marco Burattie, Federico Della Crocea,b affabio.salassa, [email protected] Dipartimento di Ingegneria Gestionale e della Produzione, Politecnico di Torino (Italy) bCNR, IEIIT, Torino, Italy [email protected] Canada Excellence Research Chair in Data Science for Real-time Decision-making, Ecole´ Polytechnique de Montr´eal(Canada) [email protected] DICATAM, Universit`adegli Studi di Brescia (Italy) [email protected] Dipartimento di Matematica e Informatica, Universit`adegli Studi di Perugia (Italy) Abstract The Oberwolfach Problem OP (F ) { posed by Gerhard Ringel in 1967 { is a paradigmatic Combinatorial Design problem asking whether the com- plete graph Kv decomposes into edge-disjoint copies of a 2-regular graph F of order v. In this paper, we provide all the necessary equipment to generate solutions to OP (F ) for relatively small orders by using the so-called differ- ence methods. From the theoretical standpoint, we present new insights on the combinatorial structures involved in the solution of the problem. Com- putationally, we provide a full recipe whose base ingredients are advanced optimization models and tailored algorithms. This algorithmic arsenal can solve the OP (F ) for all possible orders up to 60 with the modest comput- ing resources of a personal computer. The new 20 orders, from 41 to 60, encompass 241200 instances of the Oberwolfach Problem, which is 22 times greater than those solved in previous contributions. arXiv:1903.12112v4 [math.CO] 10 Nov 2020 Keywords: Combinatorics, Graph Factorization, Integer Programming, Combinatorial Design, Oberwolfach Problem, Constraint Programming ∗Corresponding author. -
Hamilton-Waterloo Problem with Triangle and C9 Factors
Michigan Technological University Digital Commons @ Michigan Tech Dissertations, Master's Theses and Master's Dissertations, Master's Theses and Master's Reports - Open Reports 2011 Hamilton-Waterloo problem with triangle and C9 factors David C. Kamin Michigan Technological University Follow this and additional works at: https://digitalcommons.mtu.edu/etds Part of the Mathematics Commons Copyright 2011 David C. Kamin Recommended Citation Kamin, David C., "Hamilton-Waterloo problem with triangle and C9 factors", Master's Thesis, Michigan Technological University, 2011. https://doi.org/10.37099/mtu.dc.etds/207 Follow this and additional works at: https://digitalcommons.mtu.edu/etds Part of the Mathematics Commons THE HAMILTON-WATERLOO PROBLEM WITH TRIANGLE AND C9 FACTORS By David C. Kamin A THESIS Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Mathematical Sciences MICHIGAN TECHNOLOGICAL UNIVERSITY 2011 c 2011 David C. Kamin This thesis, “The Hamilton-Waterloo Problem with Triangle and C9 factors”, is hereby approved in partial fulfillment of the requirements for the Degree of MASTER OF SCI- ENCE IN MATHEMATICAL SCIENCES. Department of Mathematical Sciences Signatures: Thesis Advisor Dr. Melissa Keranen Co-Advisor Dr. Sibel Ozkan¨ Department Chair Dr. Mark Gockenbach Date Contents List of Figures . vii Acknowledgements . ix Abstract . xi 1 Introduction . 1 1.1 Graph Decomposition . 1 1.2 The Oberwolfach Problem . 3 1.3 The Hamilton-Waterloo Problem . 4 1.4 Graph Theory . 6 1.5 The Mathematical Connection . 8 1.6 Previous Results . 9 1.7 Design Theory . 11 2 Generating URDs for fixed r, s .......................... 13 3 When m = 3 and n = 9 ............................. -
Mathematisches Forschungsinstitut Oberwolfach
MATHEMATISCHES FORSCHUNGSINSTITUT OBERWOLFACH Tag u n 9 s b e r i c ht ")1/1984 Graphentheorie 8.7. bis 14.7.1984 Die Tagung fand unter der Leitung von' Herrn' G. Ringel (Santa Cruz, California) und Herrn W. Mader (Hannover) statt. Mehr als 40 Teilnehmer aus 13 Ländern (Australien, 'VR " China, CSSR, Dänemark, England, Frankreich, ISI;'ael,.·· Jugoslawien, Kanada, Niederlande, Ungarn, USA) haben über ihre n~uesten Ergebnisse in verschiedenen Gebieten der Graphentheorie b~richtet. In einer von Herrn P. Er~' dös geleiteten Problemsitzung wurden offene Probleme' vorgestellt und diskutiert. Im besonderen wurden Resultate aus den folgenden Teil- ~ gebieten der Graphentheorie vorgetragen: Färbungsprobleme, (Kanten- bzw. Knotenfärbunge~/·Larid~· ka.rten) , zusammennangsprobleme, Unendliche. Graphen, Einbettungen.von Graphen in verschiedene Fiächen, (Mini malbasen, Kantenkreuzungen), Partitionen (Turan-Graphen, Ramsey-Theorie usw.), . ~. .. '\ Wege und Kreise in Graphen. Darüber hinaus wurden graphentheoretische Methoden be- © nutzt im Zusammenhang mit Matroiden, Netzwerken, Opti mierungsproblemen,' physikalischen Problemen, Lateini schen Rechtecken. Als spezielle Ergebnisse seien noch die Fortschritte im "Oberwolfacher Problem" erwähnt sowie die Lösung der seit 1890 offenen Imperium-Vermutung von Heawood. Die Grundidee zu dieser Lösung ist in der abgebildeten Land karte mit 19 paarweise benachbarten, aus jeweils 4 ge trennten Ländern bestehenden Staaten angedeutet. © - 3 - vortragsauszüge- R. AHARONI: Matchings in infinite graphs A criterion is present~d for a graph· at" any cardinality to passess a perfeet matc~ing. B. ALSPACH: Same new results on. the Oberwolfach problem Let F(11,12, •.• ,lr) IG denate an isomorphi~ factorization of G- into 2-factors each of which is compased of cycles af lengths l1,12, ••• ,lr. If a~l the li'5 are the same value 1, write F(l»)G instead.