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ANTIMAGIC LABELING of GRAPHS Oudone Phanalasy ANTIMAGIC LABELING OF GRAPHS Oudone Phanalasy A thesis submitted in total fulfilment of the requirement for the degree of Doctor of Philosophy School of Electrical Engineering and Computer Science Faculty of Engineering and Built Environment The University of Newcastle NSW 2308, Australia August 2013 Certificate of Originality This thesis contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or writ- ten by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the University Library, being made avail- able for loan and photocopying subject to the provisions of the Copyright Act 1968. (Signed) Oudone Phanalasy i Publications Arising from This Thesis 1. M. Baˇca,M. Miller, O. Phanalasy and A. Semaniˇcov´a-Feˇnovˇc´ıkov´a,Super d-antimagic labelings of disconnected plane graphs, Acta Mathematica Sinica (English Series), 26(12) (2010), 2283-2294. 2. O. Phanalasy, M. Miller, L. Rylands and P. Lieby, On a relationship between completely separating systems and antimagic labelling of regular graphs, Lec- ture Notes in Computer Science, 6460 (2011), 238-241. 3. J. Ryan, O. Phanalasy, M. Miller and L. Rylands, On antimagic labelling for generalized web and flower graphs, Lecture Notes in Computer Science, 6460 (2011), 303-313. 4. M. Miller, O. Phanalasy and J. Ryan, All graphs have antimagic total label- ings, Electr. Notes in Discrete Mathematics, 38 (2011), 645-650. 5. L. Rylands, O. Phanalasy, J. Ryan and M. Miller, Construction for antimagic generalized web graphs, AKCE Int. J. Graphs Comb., 8(2) (2011), 141-149. 6. M. Miller, O. Phanalasy, J. Ryan and L. Rylands, Antimagicness of some families of generalized graphs, Austral. J. Combin., 53 (2012), 179-190. 7. M. Baˇca,L. Brankovic, M. Lascs´akov´a,O. Phanalasy and A. Semaniˇcov´a- Feˇnovˇc´ıkov´a,On d-antimagic labellings of plane graphs, Electr. J. Graph Theory and Applications, 1(1) (2013), 28-39. 8. M. Baˇca,M. Miller, O. Phanalasy and A. Semaniˇcov´a-Feˇnovˇc´ıkov´a,Construc- tion antimagic labelings for some families of regular graphs, J. Algorithms and Computation, 44 (2013), 1-7. 9. M. Miller, O. Phanalasy, J. Ryan and L. Rylands, Sparse graphs with vertex antimagic edge labelings, AKCE Int. J. Graph Comb., 10(2) (2013), 193-198. ii 10. S. Arumugam, M. Miller, O. Phanalasy and J. Ryan, Antimagic labelling of generalized pyramid graphs, Acta Mathematica Sinica (English Series), In press. 11. L. Rylands, O. Phanalasy, J. Ryan and M. Miller, An application of com- pletely separating systems to graph labeling, Lecture Notes in Computer Sci- ence, Accepted. 12. M. Miller, O. Phanalasy, J. Ryan and L. Rylands, A note on antimagic label- ings of trees, Bulletin of ICA, Submitted. 13. O. Phanalasy, Antimagic labelling of generalized sausage graphs, Utilitas Math., Submitted. 14. J. W. Daykin, C. S. Iliopoulos, M. Miller and O. Phanalasy, Antimagicness of generalized corona and snowflake graphs, JCMCC, Submitted. 15. M. Baˇca,M. Miller, O. Phanalasy, J. Ryan, A. Semaniˇcov´a-Feˇnovˇc´ıkov´aand Anita A. Sillasen, Totally antimagic graphs, In preparation. 16. M. Baˇca,O. Phanalasy, J. Ryan and A. Semaniˇcov´a-Feˇnovˇc´ıkov´a,Antimagic labelling of join graphs, In preparation. 17. M. Miller, O. Phanalasy and J. Ryan, All graphs have vertex antimagic and edge antimagic total labelings, In preparation. 18. D. Buset, M. Miller, O. Phanalasy and J. Ryan, Antimagicness for family of generalized antiprism graphs, In preparation. iii Further Publications Produced During my Candidature 1. O. Phanalasy, I. T. Roberts and L. Rylands, Covering separating systems and application to search theory, Austral. J. Combin., 45 (2009), 3-14. 2. O. Phanalasy, M. Miller, C. S. Iliopoulos, S. P. Pissis and E. Vaezpour, Con- struction of antimagic labeling for the Cartesian product of regular graphs, Mathematics in Computer Science, 5(1) (2011), 81-87. iv Acknowledgements I would like to thank my supervisor Professor Mirka Miller for giving me a chance to do research. Her invaluable guidance, encouragement, constant support and friendship throughout my PhD candidature have combined to make my research experience worthwhile. I would like to thank my co-supervisor Dr Joe Ryan for his constant and invalu- able support and many enjoyable and useful discussions during my candidature. Throughout my candidature he was always there to help me not only with research but also non-academic problems. My special thanks go to my external supervisor Associate Professor Leanne Rylands for her friendship. I am thankful that our earlier collaboration could continue during my PhD candidature. I have learned a lot from my external supervisor Dr Andrea Semaniˇcov´a-Feˇnovˇc´ıkov´a.Working on problems together with Andrea and Professor Martin Baˇcacontributed greatly to my training as a researcher. During my PhD candidature I have benefitted from discussions with Professor Brian Alspach, Professor Subramanian Arumugam, Dr Paulette Lieby and Dr Yuqing Lin. In particular, I would like to thank Dr Ian Roberts who as my M.Sc. supervisor first inspired me to become a researcher and introduced me to combinatorics of finite sets. My thanks also go to the School of Electrical Engineering and Computer Science, the University of Newcastle and the Graduate Studies Office who provided excellent v facilities and research environment. No acknowledgement would be complete without a recognition of the contributions of my sisters, brothers and especially my wife Kipmala Xayavong and my daughters, Donemala Phanalasy, Phoutmala Phanalasy, Donesanty Phanalasy and Oumala Phanamaly for their love and encouragement. I could not have finished my PhD studies without their continuous support. vi Dedication To the memory of my parents Noulith and Thongdam Phanalasy, and my mentor and benefactor Professor Beth Southwell. vii Contents Declaration i Publications Arising from This Thesis ii Further Publications Produced During my Candidature iv Acknowledgements v Dedication vii Abstract 1 Chapter 1 Introduction 2 Chapter 2 Basic Terminology in Graph Theory 6 2.1 Operations on Graphs . .7 I EDGE LABELING 10 Chapter 3 Basic Concepts and Literature Review 11 viii 3.1 Completely Separating Systems . 11 3.2 Antimagic Labeling of Graphs . 14 Chapter 4 Application of Combinatorics to Antimagic Labeling of Graphs 17 4.1 Relationship between Completely Separating Systems and Labeling of Graphs . 17 4.2 Modification of Completely Separating Separating Systems . 21 4.2.1 Edge Deletion with no Isolated Vertex . 21 4.2.2 Edge Switching . 21 4.2.3 Splitting of Roberts' Construction . 23 4.3 Generalized Antiprism Graphs . 24 4.4 Conclusion . 41 Chapter 5 Antimagicness of some Families of Graphs 43 5.1 Generalized Web and Flower Graphs . 45 5.2 Generalized Sausage Graphs . 55 5.3 Generalized Corona and Snowflake Graphs . 61 5.4 Join of Graphs . 69 5.5 Trees and Unicyclic Graphs . 78 5.6 Conclusion . 83 II TOTAL LABELING 85 Chapter 6 Basic Concepts and Literature Review 86 ix 6.1 Total Labeling . 86 6.2 d-Antimagic Labeling . 89 Chapter 7 Total and Totally Antimagic Labeling of Graphs 91 7.1 Total Labeling of Graphs . 91 7.1.1 Vertex Antimagic Total Labeling . 91 7.1.2 Edge Antimagic Total Labeling . 96 7.2 Totally Antimagic Total Labeling of Graphs . 99 7.2.1 Join Graph . 100 7.2.2 Corona Graphs . 108 7.2.3 Union of Graphs . 112 7.3 Conclusion . 114 Chapter 8 Antimagic Labeling of Plane Graphs 115 8.1 Super d-Antimagic Labeling of Type (1; 1; 0) . 115 8.1.1 Edge Antimagic Labeling of Paths . 115 8.1.2 Partitions with Determined Differences . 117 8.1.3 d-Antimagic Labeling for Certain Families of Plane Graphs . 119 8.2 Super d-Antimagic Labeling of Type (1; 1; 1) . 129 8.2.1 Super 1-Antimagic Labeling of Type (1; 1; 1) . 129 8.2.2 Super d-Antimagic Labeling of Type (1; 1; 1) . 134 8.2.3 Some Classes of Plane Graphs . 144 x 8.3 Conclusion . 146 Chapter 9 Summary 147 9.1 Open Problems . 148 References 150 xi List of Tables n Sk 4.1 Value (i; t) corresponding to value m for A E(Gj), 1 k n. 29 G − j=1 ≤ ≤ n Sk 4.2 Value (i; t) corresponding to value m for T E(Gj), 1 k n, G− j=1 ≤ ≤ n even. 36 n Sk 4.3 Value (i; t) corresponding to value m for T E(Gj), 1 k n, G− j=1 ≤ ≤ n odd................................... 40 xii List of Figures 2.1 The Cartesian product C4 P3.....................8 × 2.2 The join P2 + C3.............................8 2.3 The graphs G and G e.........................8 − 2.4 The corona graph P3 C3........................9 3.1 Antimagic labeling of the cycle C6 and the complete graph K4.... 15 4.1 The graph G = (V; E) with edge labeling. 19 4.2 The graph B(6; 4) with antimagic edge labeling. 20 4.3 The graph G D with antimagic edge labeling. 22 − 4.4 The graph K4 with antimagic edge labeling. 22 4.5 An array CSS obtained by edge switching and corresponding graph 2 2 K4 ./ G with antimagic edge labeling. 23 4.6 The array (CSS) L1 and graph G(L1) with antimagic edge labeling. 24 4.7 The array (CSS) L2 and graph G(L2) with antimagic edge labeling. 24 4.8 The generalized antiprism A3 with antimagic edge labeling. 26 C4 4.9 The generalized toroidal antiprism T 4 with antimagic edge labeling. 34 C4 xiii 5.1 Illustration of the construction of the generalized flower graph FL(Kp; m; n; p), for p 2, m 2, n 3, and m even.
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