Lecture Notes m Mathematics

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1 Lecture Notes m Mathematics For information about Vols , please contact your bookseller or Springer-Verlag. Vol. 401: M. F. Atiyah, Elliptic Operators and Compact Groups. V, 93 pages Vol. 402: M. Waldschmidt, Nombres Transcendants. VIII, 277 pages Vol. 403: Combinatorial Mathematics. Proceedings Edited by D. A. Holton. VIII, 148 pages Vol. 404: Theorie du Potentiel et Analyse Harmonique. Edite par J. Faraut. V, 245 pages Vol. 405: K. J. Devlin and H. Johnsbraten, The Souslin Problem. VIII, 132 pages Vol. 406: Graphs and Combinatorics. Proceedings Edited by R. A. Sari and F. Harary. VIII, 355 pages Vol. 407: P. Berthelot, Cohomologie Cristalline des Schemas de Caracteristique p > o. II, 604 pages Vol. 408: J. Wermer, Potential Theory. VIII, 146 pages Vol. 409: Fonctions de Plusieurs Vanables Complexes, Semina1re Fran~ois Norguet XIII, 612 pages Vol. 410: S<iminaire Pierre Lelong (Analyse) Annee VI, 181 pages Vol. 411 : Hypergraph Seminar. Ohio State Umversity, Edited by C. 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2 Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 622 Combinatorial Mathematics V Proceedings of the Fifth Australian Conference, Held at the Royal Melbourne Institute of Technology, August 24-26, 1976 Edited by C. H. C. Little Springer-Verlag Berlin Heidelberg NewYork 1977

3 Editor Charles H. C. Little Royal Melbourne Institute of Technology Department of Mathematics and Computer Science Box No V. G. P. O. Melbourne, Victoria Australia 3001 AMS Subject Classifications (1970): 05-00,05 B20, 15A36, 62 K99 ISBN Springer-Verlag Berlin Heidelberg New York ISBN Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. by Springer-Verlag Berlin Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2140/

4 TABLE OF CONTENTS INVITED ADDRESSES Wilfried Imrich Subgroup theorems and graphs. 1 R.W. Robinson Counting unlabeled acyclic digraphs. 28 CONTRIBUTED PAPERS T.H. Andres and R.C. Stanton Golay Sequences. 44 Adrian Baddeley The knotted hexagon. 55 I.R. Beaman and W.D. Wallis On skew Room squares. 61 Peter Eades Some new constructions for orthogonal designs using circulants Peter Eades, Jennifer Seberry Wallis and Nicholas Wormald A note on asymptotic existence results for orthogonal designs. 76 C. Godsil, D.A. Holton and B. McKay The spectrum of a graph. 91 Katherine Heinrich Latin squares composed of four disjoint subsquares. 118 D.A. Holton, K.C. Stacey and K.L. McAvaney The semi-stability of lexicographic products. 128 Charles H.C. Little On rings of circuits in planar graphs. 133 Sheila Oates Macdonald Sum-free sets in loops. 141 K. L. HcAvaney, D.A. HoIton and Douglas D. Grant Groups with stable graphs. 148 M.J. Pelling and D.G. Rogers A problem in the design of electrical circuits, a generalized subadditive inequality and the recurrence relation. j(n, m) = j( m) + j( [n;l], m) + j(n, m-i l ). 153 Peter J. Robinson Orthogonal designs in order

5 IV D.G. Rogers A Schroder triangle: three combinatorial problems. 175 Jennifer A. Scoble A combinatorial approach to map theory. Anne Penfold Street 197 On quasi-multiple designs Marta Sved A generalisation of the binomial coefficients. 209

6 PREFACE These are the proceedings of the Fifth Australian Conference on Combinatorial Mathematics. The conference, which was held from 24th to 26th August 1976 at the Royal Melbourne Institute of Technology, attracted forty-six participants and there were thirty-one talks. We were fortunate to have two distinguished invited addresses, one by Professor W. Imrich of Leoben, Austria, and the other by Professor R.W. Robinson of Newcastle, Australia. This volume contains the text of their talks as well as those of a number of contributed papers. There are, as always, too many people who made significant contributions to the smooth running and success of the conference for them to be thanked individually here. Particular thanks though, should be extended to all those who chaired sessions and refereed papers. The administration of the R.M.I.T. also deserves thanks for providing the facilities for the conference. I especially wish to thank Mr. R. Vasudeva, the head of the Department of Mathematics and Computer Science, who gave his full backing and much encouragement to the conference. Dr. D.A. Holton also provided invaluable advice, assisted with the editing of the Proceedings and generously provided transportation for the participants in his car. Finally I would like to thank Miss J. Poore and Mrs. I. Leitmanis whose aid was invaluable in preparing circulars and assisting generally in the administration of the conference. C.H.C. Little

7 PARTICIPANTS R. Antill A. Baddeley I. Beaman R. Casse K. Chidzey B.M. Corrigan C. Dibley A. Dobson P. Eades S.R. Giles D. Glynn C. Godsil J. Hammer A. Hartman K. Heinrich D.A. Holton R. Hubbard W. Imrich C. Kohout C.H.C. Little S.O. Macdonald K.L. McAvaney B. McKay E.J. Morgan P. O'Halloran P. Pattison P.J. Robinson R.W. Robinson D.G. Rogers D. Row J.A. ScobIe E. Sonenberg G. Southern K.C. Stacey R.G. Stanton J. Stillwell A.P. Street M. Sved E. Szekeres Australian National University University of Adelaide Elwood, Victoria Griffith University Australian National University South Kingsville, Victoria University of Adelaide University of Sydney Monash University Queensland Institute of Technology Montanuniversitaet, Leoben, Austria Williamstown, Victoria Royal Melbourne Institute of Technology Gordon Institute of Technology Canberra College of Advanced Education Australian National University University of Western Australia University of Tasmania Royal Melbourne Institute of Technology Monash University Burwood State College University of Manitoba, Canada Monash University University of Adelaide Macquarie University