Gerhard Ringel Rainer Bodendiek . Rudolf Henn (Eds.)

Topics in Combinatorics and Essays in Honour of Gerhard Ringel

With 178 Figures

Physica-Verlag Heidelberg Professor Dr. Rainer Bodendiek Institut flir Mathematik und ihre Didaktik Padagogische Hochschule Kiel OlshausenstraBe 75 2300 Kiel, FRG

Professor Dr. RudolfHenn (died 1989) Institut flir Statistik und Mathematische Wirtschaftstheorie Universitat Karlsruhe Rechenzentrum, Zirkel 2 Postfach 6980 7500 Karlsruhe, FRG

ISBN-13: 978-3-642-46910-7 e-ISBN-13: 978-3-642-46908-4 DOl: 10.1007/978-3-642-46908-4 CIP-Titelaufnahme der Deutschen Bibliothek Topics in combinatorics and graph theory: essays in honour of Gerhard RingellRainer Bodendiek; Rudolf Henn (eds.). - Heidelberg: Physica-VerJ., 1990 ISBN-13: 978-3-642-46910-7 NE: Bodendiek, Rainer [Hrsg.); Ringel, Gerhard: Festschrift

This work is subject to copyright. All rights are reserved, whether the whole orpart ofthe material is con• cerned, specifically the rights oftranslation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication ofthis pUblication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version ofJune 24,1985, and a copyright fee must always be paid. Violations fall under the pro• secution act of the German Copyright Law. © Physica-VerJag Heidelberg 1990 Softcover reprint of the hardcover 1st edition 1990 The use ofregistered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

7120/7130-543210 Preface

Graph theory and combinatorics are two different areas of discrete mathe• matics. Although graph theory is intimately related to combinatorics and it is not always easy to decide whether certain contributions belong to graph theory or combinatorics. it_is obvious that graph theory is not a part of combinatorics. This means:that these ~WQ areas are two independent members of discrete mathematfcs.' Furthermore,.it is nearly superfluous to mention that there are a lot of applications of graph theory to some areas of physics, chemistry, commuciation science, computer technology, operatio• nal research, psychology, linguistics, economics and a great number of branches of mathematics including group theory, matrix theory, numerical analysis, probability, topology and combinatorics.

Therefore, it is not very surprising that the graph theory has got a fan• tastic development during the last thirty years. The number of graph theorists and of papers dealing with graph theoretical topics increases enormously. It is allowed to mention the graph theorists Klaus Wagner, Gerhard Ringel and Horst Sachs as pioneers of graph theory in Germany. Since one of them, Gerhard Ringel, has become 70 last year (1989) , we chose the Thursday during the graph theory meeting, held at Oberwolfach from 3-9 June 1990, to honour the work of Gerhard Ringel. About 50 mathematicians from 15 countries participated in lectures and discussions on graph theory related to Ringel's work.

Gerhard Ringel is among the most quoted graph theorists. He has developed concepts and methods of great originality, made a lot of fundamental discoveries and conjectures, and has seen the necessity to introduce graph theory in economics. In order to manage a fruitful cooperation between economics and graph theory Gerhard Ringel worked together with the economist Rudolf Henn, the co-editor of this volume. Rudolf Henn died in November 1989. Although he was not a mathematician he became a great friend of the graph theory and introduced various graph theoretical methods in economics and wrote several books dealing with a lot of topics of graph theory in economics. VI

Although he was seriously ill,he managed to settle the arising problems in a wonderful manner such that it was a great honour to me to work together with Rudolf Henn.

This volume contains 86 contributions in honour of Gerhard Ringel given by 128 authors from 22 countries. I wish to thank all the contributors. I also thank Gerda Schmidt and Dr. Claus Thies who have helped preparing and typing the manuscripts. Professor Heiko Harborth is thanked for giving the appreciation of Gerhard Ringel. Finally, we are especially grateful to Dr. Werner A. MOller from Physica-Verlag for his cooperation in all aspects of the production of this book.

Rainer Bodendiek Contents

GERHARD RINGEL - SEVENTY XIII H. Harborth ON THE PROBLEM OF RELATIVE COMPONENTS OF MINIMAL GRAPHS R. Bodendiek / K. Wagner IRREGULAR ASSIGNMENTS AND TWO PROBLEMS a la RINGEL 29 M. Aigner / E. Triesch A RECURSIVE BOUND FOR THE NUMBER OF COMPLETE K-SUBGRAPHS OF A GRAPH 37 R. Ahlswede / N. Cai / Z. Zhang ONE-FACTORIZATIONS OF TENSOR PRODUCTS OF GRAPHS 41 B. Alspach / J.C. George NON-COMMUTATIVE GEOMETRY AND GRAPHS 47 J. Andre THE COMPLEXITY OF THE GRAPH EMBEDDING PROBLEM 59 D. Archdeacon HELLY THEOREMS FOR DISMANTLABLE GRAPHS AND PSEUDO-MODULAR GRAPHS 65 H.-J. Bandelt / H.M. Mulder ON THE LEVEL-ORIENTED TWO-DIMENSIONAL PACKING WITH ROTATION 73 OF THE RECTANGLES G. B!ir ON PLANAR TILINGS WITH FINITELY MANY SORTS OF TILES 79 K. Bezdek EXAMPLES OF SPACE-TILING POLYHEDRA RELATED TO HILBERTS PROBLEM 18. 87 QUESTION 2 A. Bezdek / W. Kuperberg THE HISTORICAL BACKGROUND TO GERHARD RINGEL's WORK 93 N.L. Biggs / E.K. Lloyd / R.J. Wilson AROUND THREE LEMMAS IN HAMILTONIAN GRAPH THEORY 101 D. Bauer / H.J. Broersma / H.J. Veldman A NOTE ON METRIC PROPERTIES OF INFINITE GRAPHS 111 T. BOhme AUTOMOTPHISM GROUPS OF DIRECTED CAYLEY GRAPHS 117 U. Baumann / M. Lesch / I. Schmeichel TRIANGULAR EMBEDDINGS OF TENSOR PRODUCTS OF GRAPHS 129 A. Bouchet / B. Mohar VIII

COMPUTING LIGHT EDGES IN PLANAR GRAPHS 137 O.V. Borodin ON THE DOMINATION PROBLEM FOR BIPARTITE GRAPHS 145 A. Brandstadt POLYHEDRAL MAPS WITH FEW EDGES 153 U. Brehm AUT Gm,n FOR THE HASSE GRAPH Gm,n OF THE SUBWORD POSET Bm,n 163 OF AN m-ARY CYCLIC WORD OF LENGTH n G. Burosch / J.-M. Laborde STATUS OF GRACEFUL TREE CONJECTURE IN 1989 175 I. Cahit EMBEDDED GRAPHS, FACIAL COLORINGS, AND DOUBLE CYCLE COVERS 185 P.A. Catlin ON PERIPHERAL VERTICES IN GRAPHS 193 G. Chartrand / G. Johns / O.R. Oellermann THE VERTEX-DEGREES OF STEINER MINIMAL TREES IN MINKOWSKI PLANES 201 D. Cieslik UNFOLDING WEIGHTED CONCENSUS ORDERS 207 INTO CONSISTENT NUMERICAL SCALES J. Czyzowicz / A. Pelc / I. Rival FORBIDDEN ORDERED SUBGRAPHS 219 P. Damaschke ON NORMAL TOURNAMENTS WITH THE LEAST NUMBER OF 3-CYCLES 231 D.C. Demaria / G.M. Gianella TWO-IRREGULAR GRAPHS 239 R.J. Faudree / R.J. Gould / M.S. Jacobsen / R.H. Schelp CELL COMPLEXES AND LOWER BOUNDS IN COMPUTATIONAL GEOMETRY 249 T. Fischer CHARACTERIZING DIRECTED POSTMAN TOURS 257 H. Fleischner / E. Wenger SOME PROPERTIES OF "ALMOST ALL" FUNCTIONS FROM Pk 263 I.D. Giudjenov COMPOSITION OF FACETS OF THE CLIQUE PARTITIONING POLYTOPE 271 M. Gr5tschel / Y. Wakabayashi OPTIMAL EDGE-NUMBERING OF BINARY TREES 285 N. GrUnwald IX

ON INDEPENDENT VERTICES AND EDGES OF A GRAPH 291 I. Gutman THE OUTERTHICKNESS &OUTERCOARSENESS OF GRAPHS 297 I. THE &THE n-CUBE R.K. Guy I R.J. Nowakowski ON SOME GRAPHIC ASPECTS OF ADDITION THEOREMS 311 Y.O. Hamidoune ON THE CIRCUMFERENCE OF REGULAR POLYHEDRAL GRAPHS 319 J. Harant I H. Walther LONGEST CYCLES IN CIRCULANT GRAPHS 331 W. Harnau I D. Jordan SPANNING TREES OF THE COMPLETE BIPARTITE GRAPH 339 N. Hartsfield I J.S. Werth A COMBINATORIAL THEOREM WHICH IS RELATED TO THE INVARIANCE OF 347 THE SEPARATING SET FOR THE PLANE E. Harzheim ON CERTAIN TREES IN HYPERCUBES 353 I. Havel EXTENDING HALL1S THEOREM 359 A.J.W. Hilton I P.O. Johnson Jr. ON THE COUPLING CONDITION AND HAMIL TONICITY 373 C. Hoede I H.J. Smit TRANSVERSALS AND MATROIDS 381 P. Hor&k CLASSIFICATION AND CONSTRUCTION OF GEODETIC BLOCK 391 WITH DIAMETER TWO M. Jingzhong GRAPH DISTANCES AND SIMILARITY 397 F. Kaden WITT RINGS AND SEMI ORDERINGS OF PLANAR TERNARY RINGS 405 F.B. Kalhoff LINEAR INEQUALITIES DESCRIBING THE CLASS OF INTERSECTING SPERNER 413 FAMILIES OF SUBSETS. I G.O.H. Katona I G. Schild INTEGRAL DRAWINGS OF THE COMPLETE GRAPH K6 421 A. Kemnitz x

ON CERTAIN BINOMIAL SUMS 431 W. Klotz COLOURING OF SPIDER GRAPHS 435 M. Koebe A LAS-VERGNAS TYPE THEOREM FOR TREES 443 I. Krasikov / Y. Roditty QUICK GOSSIPING BY MULTI-TELEGRAPHS 451 R. Labahn / I. Warnke NUMBERINGS ON GRAPHS HAVING SMALL EDGE WEIGHTS 459 R. Lang ON VERTEXMINIMAL GRAPHS WITH RADIUS r AND CONNECTIVITY 2m 471 G. Lassmann EMBEDDING SCHEMES AND THE JORDAN CURVE THEOREM 479 C.H.C. Little / A. Vince SUBGRAPH PACKING - A SURVEY 491 M. Loebl / S. Poljak ON THE RADIUS OF RANDOM SUBGRAPHS OF THE n-CUBE 505 K. Mahrhold / K. Weber A RESULT IN COMBINATORIAL MATROID THEORY 513 D. Marcu ON GRAPHS EMBEQDABLE WITH SHORT FACES 519 R. Nedela / MSkoviera ON CYCLIC REPRESENTATIONS OF TRIPLES BY PAIRS 531 J. Novak ON THE STEINER PERIPHERY AND STEINER ECCENTRICITY OF A GRAPH 541 O.R. Oellermann / H.C. Swart CYCLES CONTAINING THREE CONSECUTIVE EDGES 549 IN 2k-EDGE-CONNECTED GRAPHS H. Okamura GRAPH DISTANCE AND EUCLIDEAN DISTANCE ON THE GRID 555 J. Pach / R. Pollack / J. Spencer ABOUT THE COMPLEXITY OF SOME HOMOMORPHISM PROBLEMS ON GRAPHS 561 A. Quilliot ON AN INEQUALITY OF SPERNER 569 A. Rausche COUNTING PERFECT MATCHINGS IN LATTICE GRAPHS 577 H. Sachs XI

GENYS -vMINIMAL EDGES AND KURATOWSKI SUBGRAPHS OF A GRAPH 585 J. Sir~n FROM TREE PATH-FACTORS AND DOUBLY EXPONENTIAL SEQUENCES 595 TO A BINOMIAL INEQUALITY Z. Skupien A CHARACTERIZATION OF POINT-COLOUR-SYMMETRIC HYPERGRAPHS 605 S. Sze-Chin A LINEAR ALGORITHM FOR THE PATHWIDTH OF TREES 613 P. Scheffler THE TIME TRAVELLING PROBLEM 621 G. Schild AN APERIODIC TRIPLE OF PROTOTILES 627 P. Schmitt REPRESENTATION OF GRAPHS BY INTEGERS 635 R. Schnabel SPECIAL SYSTEMS OF LINEAR EQUATIONS AND GRAPHS OF CONVEX 641 POLYTOPES W. SchOne ON 2-EMBEDDABLE GRAPHS 651 H. Schumacher ON AN APPLICATION OF THE BOOLEAN DIFFERENTIAL CALCULUS TO 663 DIGITAL SYSTEM THEORY B. Stiefel EQUIAREAL SETS IN Rd 671 G.W. Teumer ON THE PIAGET GRAPH 679 C. Thies ON A CHARACTERIZATION OF CLOSURE OPERATORS BY IDENTITIES ON 685 SEMI GROUPS R. Thron / J. Koppitz SYMMETRIES OF GROUP-TRIANGULATIONS 693 H.-J. Voss EXPERIMENTAL MATHEMATICS - TESSELATIONS OF CONVEX POLYGONS 713 IN A HEXAGONAL LATTICE G. Walther XII

DOMINATION IN CUBIC GRAPHS 727 B. Zelinka A GENERALIZATION OF THE BODENDIEK CONJECTURE ABOUT GRACEFUL GRAPHS 737 C. Zhi-Zeng A SPARSE CALLAI-WITT THEOREM 747 H.J. Prornel I B. Voigt EDGES WITH AT MOST ONE CROSSING IN DRAWINGS OF THE COMPLETE GRAPH 757 H. Harborth / I. Mengersen LONG CYCLES IN GRAPHS WITH MODERATE CONNECTIVITY 765 H.A. Jung INDEPENDENT COVERS IN PLANE GRAPHS 779 M.M. Syslo LIST OF CONTRIBUTORS 786 Gerhard Ringel - Seventy

October 28, 1989, was the seventieth return of Gerhard Ringel's birthday. On this occasion around seventy mathema• ticians have submitted articles for this volume to honor Gerhard Ringel who himself has published around seventy mathe• matical papers. So there are reasons enough to celebrate seventy.

In the mathematical world of Graph Theory and Combinatorics Gerhard Ringel is a well-known mathematician. Since nearly fourty years he has solved some old conjectures, mainly in topological Graph Theory, and he created several nice new problems which stimulated many other people working in Graph Theory.

Gerhard Ringel was born in GroBdorf near Braunau in Austria. After his attendance at the gymnasium of Braunau he had to serve in the German army since 1940. Because of a heart-defect he soon could interrupt his military service for two years, and he studied at the Karls university of Prague. Then, how• ever, more than seven years followed where he was soldier in the German army, and then a prisoner-of-war in the Soviet Union. At the end of 1949 he could continue to study mathema• tics, now at the University of Bonn. He received his doctorate in 1951, two years later the Habilitation, and he continued at Bonn as a junior faculty member.

In 1960 he moved to the Free University of Berlin as Professor of Mathematics. Also in Berlin, in 1966 he became an Ordinar• ius and Director of the Mathematical Institute. Within the next four years visiting professorships in Rome (1966), Santa Cruz (1967/68), and Calgary (1969) were included. In 1970 he left Berlin and followed the offered permanent position at the University of California, Santa Cruz, to be the successor of his friend and coauthor J.W.T. Youngs. XIV

In Santa Cruz Gerhard Ringel lives together with his wife, Isolde, in a nice house just at the shore of the Pacific Ocean. From his first marriage he has three children, Gerhard, Ingrid, and Renate.

Gerhard Ringel has several sporting hobbies. In his Institute at Berlin regularly soccer was played. In California he became a surfer. He plays chess, and still today he nearly every day plays tennis. Moreover, he collects butterflies, and for this reason the Ringels travel to exotic tropical countries, such as Costa Rica, Ecuador, or Indonesia.

Still today Gerhard Ringel serves the University of Santa Cruz as Professor. He was Head of Department for more than twelve years. His main research interests are map coloring and ernbed• dings of graphs into surfaces. These topics each year are offered by him in a combinatorics class. So far he had six doctoral students, and three of them have published several common papers with him.

In 1983 Gerhard Ringel was awarded an Honorary Doctorate by the University Fridericiana at Karlsruhe. He has written three books, one of which has been translated into Russian. The title of the forthcoming text book "Pearls in Graph Theory" gives an impression of Ringel's mental attitude to Graph Theo• ry. Gerhard Ringel is member of the editorial boards of Archiv der ~~thematik, Journal of Graph Theory, and Journal of Combi• natorial Theory B (up to 1983). Since the sixtieth he was chairman, always assisted by his wife Isolde, of all Oberwol• fach conferences in Graph Theory scheduled every two or three years.

When Gerhard Ringel started in the fiftieth to work in Graph Theory there were only very few other graph theorists among the German speaking mathematicians, and now there is a large Graph Theory community.

The main mathematical contributions of Gerhard Ringel are the proof of Heawood's map color formula, the method of current graphs, the solution of Heawood's empire conjecture, a charac• terization of self-complementary graphs, some non-strechable xv arrangements of pseudolines, minimal triangulations and quad• rangulations of 2-manifolds, and results on several measure• ments of closeness of graphs to planarity, like , thick• ness, coarseness, crossing number, or splitting number. As an example, the Figure gives a perfect splitting of the complete graph K12 , and also a map of 12 mutually adjacent empires, each of two countries.

Several nice problems and conjectures created by Gerhard Rin• gel are research objects for many mathematicians, for example, the conjecture of graceful trees, a six-color problem, recently solved by a Russian mathematician, the Oberwolfach problem, or the recent conjecture, that every connected graph, excluded K2 , is antimagic, that means, there exists a labelling of the q edges with numbers 1 to q such that all sums of the labels at any vertex are different.

Gerhard Ringel's work is part of the blooming epoch of Graph Theory during the last decades. Gerhard Ringel is not an abstract mathematician, he loves Graph Theory, he likes to see things also intuitively, he prefers nice theorems and nice conjectures, he is an aesthet. If sometimes also a book "Pearls of Graph Theorists" should be written, I think, Gerhard Ringel should be one of the pearls.

Heiko Harborth, Braunschweig List of Publications

G. Ringel

1. Farbensatz fUr nichtorientierbare Flachen beliebigen Ge• schlechts. J. Reine Angew. Math. 190 (1952), 129-147. 2. Bestimmung der Maximalzahl der Nachbargebiete auf nicht• orientierbaren Flachen. Arch. Math. i (1953), 137-142. 3. tiber polyedrische Zerlegungen geschlossener Flachen. Habi• litationsschrift, Rheinische Friedrich-Wilhelms-Universi• tat Bonn, 1953. 4. Farbensatz fUr orientierbare Flachen vom Geschlecht p>O. J. Reine Angew. Math. 193 (1954), 11-38. 5. Bestinunung der Maximalzahl der Nachbargebiete auf nicht• orientierbaren Flachen. Math. Ann. 127 (1954), 181-214. 6. Lokal-regulare Zerlegungen geschlossener orientierbarer Flachen. Math. Z. ~ (1954), 484-495. 7. tiber drei kombinatorische Probleme am n-dimensionalen WUr• fel und WUrfelgitter. Ahh. Math. Sem. Univ. Hamburg 20 (1955), 10-19. 8. Wie man die geschlossenen nichtorientierbaren Flachen in moglichst wenige Dreiecke zerlegen kann. Math. Ann. 130 (1955),317-326. 9. Teilungen der Ebene durch Geraden oder topologische Gera• den. (E. Sperner zum 50. Geburtstag am 9. Dezember 1955 gewidmet.) Math. Z. 64 (1956),79-102. 10. tiber Geraden in allgemeiner Lage. Elem. Math. 12 (1957), 75-82. - 11. Farbungsprobleme auf Flachen und Graphen. Math. Monogra• phien, 2. VEB Deutscher Verlag der Wiss., Berlin, 1959, 132 pages. 12. tiber das Problem der Nachbargebiete auf orientierbaren Flachen. (Geheimrat Prof. Dr. Lothar Heffter zum 99. Ge• burtstag gewidmet.) Math. Sem. Univ. Hamburg 25 (1961), 105-127. 13. Konfigurationen und SchlieBungssatze (Zusammenfassung eines Vortrags). Wiss. Z. Martin-Luther-Univ. Halle-Witten• berg, Math.-Nat. X (1961), 116-117. 14. Das Farbungspro0lem auf orientierbaren Flachen (Zusammen• fassung eines Vortrags). Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, ~1a th . -Nat. X (1961), 117. 15. tiber eine Klasseneinteilung der zweiecklosen Graphen. Arch. Math. ~ (1961), 231-237. 16. Selbstkomplementare Graphen. (Hans Peters son zum 60. Ge• burtstag gewidmet.) Arch. Math. 14 (1963), 354-358. XVII

17. Extremal problems in the Theory of Graphs. Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Publ. House Czechoslovak Acad. Sci., Prague 1964, 85-90. 18. Das Geschlecht des vollstandigen paaren Graphen. Ahh. Math. Sem. Univ. Hamburg 28 (1965), 139-150. 19. On the complete bipartite graph. Celebrazioni archimedee del secolo XX Simposio di didattica della matematica, Syracuse (1964). Tipografia "oderisi", Editrice Gubbio, Ita I ia , 14 1-1 45 • 20. Die toroidale Dicke des vollstandigen Graphen. (Ruth Mou• fang zum 60. Geburtstag am 10. 1. 1965 gewidmet.) Math. Z. 87 (1965), 19-26. 21. Ein Sechsfarbenproblem auf der Kugel. (Emanuel Sperner zum 60. Geburtstag gewidmet.) Ahh. Math. Sem. Univ. Hamburg 29 (1965), 107-117. 22. Der vollstandige paare Graph auf nichtorientierbaren Fla• chen. (Wolfgang Franz zum 60. Geburtstag gewidmet.) J. Reine Angew. Math. 220 (1965),88-93. 23. tiber das Geschlecht des vollstandigen Graphen. Beitrage zur Graphentheorie. Intern. Koll. M.anebach (DDR), 9.-12. f'.1ai 1967. Teubner, Leipzig 1968, 107-112. 24. A six-color problem on the . Theory of Graphs (Proc. Colloq., Tihany, 1966). Academic Press, New York 1968, 265-269. 25. (with J.W.T. Youngs) Solution of the Heawood map-coloring problem. Proc. Nat. Acad. Sci. U.S.A. 60 (1968), 438-445. 26. (with J.W.T. Youngs) Losung des Problems der Nachbargebie• teo Arch. Math. (Basel) 20 (1969), 190-201. 27. (with J.W.T. Youngs) Solution of the Heawood map-coloring problem - Case 11. J. Combinatorial Theory 1 (1969), 71-93. 28. (with J.W.T. Youngs) Solution of the Heawood map-coloring problem - Case 2. J. Combinatorial Theory 1 (1969), 342- 352. 29. (with J.W.T. Youngs) Solution of the Heawood map-coloring pro~lem - Case 8. J. Combinatorial Theory 1 (1969), 353- 363. 30. (with J.W.T. Youngs) Remarks on the Heawood conjecture. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) Academic Press, New York, 1969, 133-138. 31. (with J.\v.T. Youngs) Das Geschlecht des vollstandigen dreifarbbaren Graphen. Comment. Math. Helv. 45 (1970), 152 -158. -- 32. Genus of graphs. Combinatorial Structures and their Appli• cations (Proc. Calgary Internat. Conf., Calgary, Alta., 1969). Gordon and Breach, New York, 1970, 361-366. 33. Das Kartenfarbungsproblem. Selecta Mathematica III. Hei- XVIII

delberger TaschenbU~her, 86. Spri~ger, Berlin ~971, 27-55. 34. Kleine Verscharfung des FUnf-Farben-Satzes. Symposia Mathe• matica, Vol. V (INDAM, 1969/70). Academic Press, London 1971, 315-324. 35. Triangular embeddings of graphs. (Dedicated to the memory of J.W.T. Youngs.) Graph Theory and Applications (Proc. Conf. western Michigan Univ., Kalamazoo, Mich., 1972. Lec• ture Notes in Math., Vol. 303, Springer, Berlin 1972, 269- 281. 36. J.W.T. Youngs (1910-1970). J. Combinatorial Theory Ser. B 11 (1972), 91-93. 37. Map Color Theorem. Die Grundlehren der mathematischen Wissenschaften, Band 209. Springer, New York-Heidelberg 1974, 191 pages. 38. (with Richard K. Guy) Triangular imbedding of K -K6 • J. Combinatorial Theory Ser. B ~ (1976), 140-145.n 39. The combinatorial map color theorem. J. Graph Theory (1977), 141-155. 40. (with Mark Jungerman) The genus of the n-octahedron: regu• lar cases. J. Graph Theory ~ (1978), 69-75. 41. On the genus of the graph K XK 2 or the n-prism. Discrete Math. ~ (1977/78), 287-294~ 42. Nonexistence of graph embeddings. Theory and Applications of Graphs (Proc. Internat. Conf., Western Mich. Univ., Kalamazoo, Mich. 1976). Lecture Notes in Math. 642, Sprin• ger, Berlin 1978, 465-476. 43. (with Mark Jungerman) Minimal triangulations on orientable surfaces. Acta Math. 145 (1980), 121-154. 44. A nine color theorem for the and the . The Theory and Applications of Graphs (Kalamazoo, Mich. 1980), Wiley, New York 1981, 507-515. 45. (with Brad Jackson) Maps of m-pires on the projective plane. Discrete Math. ~ (1983), 15-20. 46. (with Brad Jackson) Colorings of circles. Amer. Math. Monthly 21 (1984), 42-49. 47. (with Brad Jackson) Heawood's empire problem. J. Combin. Theory Ser. B 38 (1985), 168-178. 48. (with Brad Jackson) The splitting number of complete bipartite graphs. Arch. Math. (Basel) 42 (1984), 178-184. 49. (with Brad Jackson) Empire maps. Graphs and Applications (Boulder, Colo., 1982). Wiley-Intersci. Publ., Wiley New York, 1985, 195- 201. 50. (with Brad Jackson) Splittings of graphs on surfaces. Graphs and APPlications (Boulder, Colo., 1982). Wiley• Intersci. Publ., Wiley New York, 1985, 203-219. 51. (with Brad Jackson) Plane constructions for graphs, net• works, and maps: measurements of planarity. Lecture Notes XIX

in Econom. and Math Syst~ms, 226, Springer, Berlin-New York, 1984, 315-324. 52. (with Brad Jackson) Coloring island maps. Bull. Austral. Math. Soc. 29 (1984), 151-165. 53. (with Brad Jackson) Solution of Heawood's empire problem in the plane. J. Reine Angew. Math. 347 (1984), 146-153. 54. (with Nora Hartsfield and Brad Jackson) The splitting number of the complete graph. Graphs Combin. 1 (1985), 311 -329. - 55. 250 Jahre Graphentheorie. (Klaus Wagner zum 75. Geburtstag gewidmet.) Graphen in Forschung und Unterricht, Fest• schrift Kiel, 1985. Franzbecker, Bad Salzdetfurth 1985, 136-152. 56. Vermutungen tiber numerierbare Graphen. Elem. Math. 41 (1986),68-74. 57. (with Nora Hartsfield) Minimal quadrangulations of non• orientable surfaces. J. Combin. Theory Sere A 50 (1989), 186-195. 58. (with Nora Hartsfield) Hamilton surfaces for the complete symmetric tripartite graph. Arch. Math. (Basel) 50 (1988), 470-473. 59. (with Nora Hartsfield) Minimal quadrangulations of orien• table surfaces. J. Combin. Theory Sere B ~ (1989), 84-95. 60. (with Nora Hartsfield) Quadrangular embeddings of the complete even k.partite graph. Discrete Math., to appear.

61. (with ~ora Hartsfield) Clean triangulations. Combinatori• ca, to appear. 62. (with Nora Hartsfield and Brad Jackson) Hamilton surfaces for the complete even symmetric bipartite graph. Discrete Math. 78 (1989),89-94. 63. (with Nora Hartsfield) Supermagic and antimagic graphs. J. Recreational Math. ~ (1989), 107-115. 64. (with Nora Hartsfield and Dragan Marusic) Nonorientable self-dual embeddings of Cayley graphs. Graphs Combin., submitted. 65. (with Nora Hartsfield) Pearls in Graph Theory. Academic Press, 350 pages, to appear. 66. (with Hanfried Lenz) A brief review on Egmont Kohler's mathematical work. Discrete Math., to appear.