Gerhard Ringel Rainer Bodendiek . Rudolf Henn (Eds.) Topics in Combinatorics and Graph Theory 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 COMPLETE GRAPH &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.