TOWARDSTHE GRACEFULTREE CONJECTURE Frank Van Busse1 A thesis submitted in conformity with the requirements for the degree of Masters of Science Graduate Department of Computer Science University of Toronto Copyright @ 2000 by Frank Van Busse1 National Library Bibliotheque nationale 1*1 of Canada du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 395 Wellington Street 395, nie Wdlingtm Ottawa ON K1A ON4 Onawa ON K1A ON4 Canada canada The author has granted a non- L'auteur a accordé une licence non exclusive Licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or seii reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/fïim, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be p~tedor otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. Abstract Tùtvards the Graceful Tree Conjecture Frank \an Busse1 Slasters of Science Graduate Depart ment of Computer Science University of Toronto 2000 In this thesis we preçent several results about graceful and near-graccful tabellings of trees. Irarious modifications of the conditions for graceful and bipartite graceful labellings are probed; we conjecture that much stronger assumptions about graceful labellings than previously considered can be appIied to al1 trees, and ive provide evidence towards these conjectures. The most significant result of this thesis concerns a strengrhening of the graceful labelling condition which requires that the centre vertes of' a tree be assigned the label 0: labellings tvhich satisfy this extra constraint are called O-centred grcrc~ful. Empirical results are presented n-hich indicate t hat non-0-centred graceful trees consticute a very small and easilj- charactcrized subset of diameter 4 trees; it is shon-n that for al1 orders rio rriernber of this subset has a O-centred graceful labelling, and that al1 other trees üf diameter 5 -1 do have one. These results are estended to O-labellings of arbitrary verlices via the notion of O-ubiquitousiy grace/ul trees; similar results (both empirical and theoretical) are obtained with respect to these. -As is-ell, some bûunds are presented for certain rclaxed graceful Iabellings of trees, ernpirical data concerning the prevalence of bipartite graceful trees is analyzed, and a n-eakened foim of bipartite labelling called localiy bipartite is investigated. Acknowledgements I would like to thank my supervisor, Mike hlolloy, who is not only a mathematician of the first rank, but an exceptional teacher and advisor. That this thesis ever made it safely into port is due primarily to his judgement and guidance- 1 would as well like to thank my second reader, Alesander Rosa, for his suggestions and advice, and of course for the Graceful Tree Conjecture itself. Throughout my research for this thesis his body of work on the GTC mas an inspiration and template. Many thanks to my friends and colleagues at the department of Computer Science here at the University of Toronto, and especially to Ioannis Papoutsakis, from whom 1 first learned about the Graceful Tree conjecture, and u-ho has ahays exhorted me to look for the essential reasons behind things. Thanks also to Joe, Leslie, Graham, and the rest of the gang at The Room 338 bistro, where bits and pieces of this thesis fell into place between discussions of quantum tunneling or art history, garnes of fuzbol, and pints, while Deep Purple played on the jukebos. 1 cannot express the gratitude I feel toward my parents, Jack and Bernadine Van Bussel, for thcir patience and support throughout my meandering career. Thanks as well to mu brother George, for his encouragement and assistance through the years. This tbesis is dedicated to my "little bl~ebell'~,Zeina Khan. iii Contents Introduction 1 1 Historical Background 4 1.1 Graph Labelling Problems .......................... 4 1.1.1 19th century and early 20th century antecedents .......... 4 1.1.2 Bandwidth and related problems .................. 6 1.1.3 Xnalogs outside of graph theory ................... 9 1.2 The Graceful Tree Conjecture ........................ 13 1.2.1 Ringel's conjecture .......................... 13 1.2. 2 General developmcnts in graceful labclling ............. 17 1.2.3 LVork towards the Graceful Tree Conjecture ............ 21 1.2.4 Tree product constructions ...................... 23 1.3 Graceful (and a few not-so-graceful) graphs ................. 28 2.4 irariations on a Graceful Theme ....................... 32 1.Near graceful labeilings ........................ 32 1.42 Relaxations of the graceful distinctness conditions ......... 35 1.4.3 Harrnonious and additive labellings ................. 36 1.1.4 Edge first labellings .......................... 39 1.4.5 Total labellings ............................ 10 1.4.6 Other types of labellings ....................... 42 1.5 The graceful tree conjecture today ...................... 42 1-5-1 Rosa and ~iraii(1995) ........................ 43 1.5.2 Aldred and Mackay (1998) ...................... 41 1.5.3 Broersma and Hoede (1999) ..................... 45 2 Relaxed Graceful Labeliings of Trees 46 2.1 -4 schematization of relaved graceful labellings ............... 46 2.2 Range-Relaxed Graceful Labellings ..................... 17 2.3 Vertes-Relaxed Graceful Labellings ..................... 54 3 Bipartite and Locally Bipartite Graceful Labellings 63 3.1 Bipartite graceful labellings ......................... 63 3.1.1 Useful fcatures of bipartite labellings ................ 63 3.1.2 Prohlcms with fully bipartite labellings ............... 66 3.2 Some empirical results ............................ 67 3.2.1 Estent of bipartite and locally bipartite graceful trees ....... 67 3.2.2 Estent of bipartite and locally bipartite graceful labellirigs .... 59 3.2.3 Other empirical results ........................ 73 3.3 -4 recursive construction for graceful tree labellings ............ 74 3.4 Some explicit constructions for trees which are not bipartite graceful ... 83 3.4.1 -4 construction for k-cornets ..................... 83 3-42 A construction for certain lobsters .................. 90 4 O-Centred and rn-Edge-Centred Gracefulness 99 4.1 O-centred and m-edge-centred graceful labellings .............. 99 4.1.1 Gracefully assigning the O label ................... 99 4.1.2 Some empirical results ........................ 101 4.1.3 Two t heorems about m-edge-centred labellings ........... 105 4.2 O-centred graceful trees of diameter 4 .................... 109 1.2.1 Trees of diameter 4 with centre degree 2 - . - . 109 4-22 Ottier trees of diameter 4 . - . - . - . 112 . 4.3 Trees that are "ubiquitously'' graceful . - . - . - 133 4.3.1 O-ubiquitously graceful trees . - . - - . - 133 4.3.2 Further empiricai results . - . 138 5 Conclusions and Future Work 141 5.1 Some open problems . - . - . 142 3.2 Working towards the graceful tree conjecture . - . 145 Appendix 149 Bibliography 154 List of Tables 3.1 Bipartite graceful and non-BG trees of order n ............... 68 3.2 Trees which are not bipartite graceful, by order n and diameter d .... 69 3.3 Locaily bipartite and bipartite graceful labellings for trees of order n . 70 3.4 Trees of order n without bipartite graceful, LB(l), or either labelling . 82 1.1 Non-O-centred and non-m-edge-centred graceful trees of order n ..... 103 4.2 Xon-bipartite graceful vs. non-O-CBG and non-m-ECBG trees ...... 105 4.3 O-ubiquitously graceful trees ......................... 134 4.4 m-edge-ubiquitously graceful t rees ...................... 138 4 5 ubiquitously graceful trees .......................... 139 vii List of Figures 1.1 Cayley diagram for the group D3....................... 5 1.2 Representations of bandwidth labellings................... 7 1.3 Tree from figure 1.2 with optimal cutwidth embedding ........... 9 1.4 Two rulers measuring 1~.3.4.5.6.7.5.9.10.11.12.13.16.17........... 12 1.5 Cjdic decomposition of K9 into a tree of size 4 ............... 14 1.6 Canonical a-valuation of a caterpillar..................... 15 1.7 The 3-cornet .................................. 16 1.8 Graceful labelling of K374........................... 16 1.9 Kotzig2sT4................................. .. 23 1.10 Tree product construction ........................... 25 1.1 1 -4 graceful labelling of a radially symmetric tree ............... 26 1.13 Graceful Iabelling for the cycle Ci ...................... 28 1.13 Graceful labelling of the Dutch 5-windmill .................. 30 1.11 Graceful labelling of the grid Pq x P5..................... 31 1.15 -4 graceful labelling of the Peterson graph .................. 32 1. 16 5-graceful IabelIing of the 3-carnet ...................... 34 1.17 Cordial labelling of the %cornet ........................ 35 1.1S Harmonious labelling of the %cornet ..................... 37 1.19 1-sequential labelling of the 3-cornet ..................... 41 1.20 h4agic Iabelling of the %cornet ........................ 41 2.1 Construction for RRG labelling of C. with n 2 or 3 (mod 4) ...... 48 2.2 Construction for Theorem 2.2 ........................ 51 2.3 RRG labelling of a tree using theorem