GEOMETRIC REPRESENTATIONS OF DISTINCT HAMILTONIAN
CIRCUITS IN COMPLETE GRAPH DECOMPOSITION
MAIZON MOHD DARUS
MASTER OF SCIENCE (MATHEMATICS)
UNIVERSITI UTARA MALAYSIA
2015
Permission to Use
In presenting this thesis in fulfilment of the requirements for a postgraduate
degree from Universiti Utara Malaysia, I agree that the Universiti Library may
make it freely available for inspection. I further agree that permission for the
copying of this thesis in any manner, in whole or in part, for scholarly purpose
may be granted by my supervisor(s) or, in their absence, by the Dean of Awang
Had Salleh Graduate School of Arts and Sciences. It is understood that any
copying or publication or use of this thesis or parts thereof for financial gain
shall not be allowed without my written permission. It is also understood that
due recognition shall be given to me and to Universiti Utara Malaysia for any
scholarly use which may be made of any material from my thesis.
Requests for permission to copy or to make other use of materials in this thesis,
in whole or in part, should be addressed to :
Dean of Awang Had Salleh Graduate School of Arts and Sciences
UUM College of Arts and Sciences
Universiti Utara Malaysia
06010 UUM Sintok
i
Abstrak
Penggambaran bagi perwakilan geometri untuk litar Hamiltonan berbeza dalam
graf lengkap diperlukan untuk mengelakkan kemiripan struktur dalam aplikasi
sebenar. Walau bagaimanapun, terdapat hanya sebilangan kecil kajian yang
mempertimbangkan penggambaran graf sedangkan ramai penyelidik memberi
tumpuan terhadap masa pengiraan. Oleh itu, kajian ini bertujuan untuk membina
satu kaedah gambaran baharu iaitu Kaedah Rama-rama Separuh (HBM) untuk
menyelesaikan senario tersebut. Bagi membina HBM, satu konsep Strategi
Sayap baharu diperkenalkan untuk mendapatkan arah dari satu bucu ke bucu
yang lain. Seterusnya, arah ini digunakan untuk memetakan bucu-bucu yang
berbeza. Bagi mendapatkan litar Hamiltonan berbeza, konsep matriks
transposisi digunakan untuk mengesan imej cermin bagi litar tersebut. Beberapa
teorem dan lema baharu dibuktikan dalam penghuraian graf lengkap kepada litar
Hamiltonan berbeza. Selanjutnya, hasil HBM ini digunakan untuk
menyenaraikan semua pilih atur 𝑛! dan beberapa teorem berkaitan dibentuk.
Kesimpulannya, kajian ini berjaya menghasilkan satu kaedah baharu untuk
menggambarkan litar Hamiltonan berbeza dalam penghuraian graf lengkap.
Kata kunci: Graf lengkap, Kaedah Rama-rama Separuh, Litar Hamilton,
Penghuraian graf
ii
Abstract
Visualization of geometric representations of distinct Hamiltonian circuits in
complete graphs is needed to avoid structures resemblance in real application.
However, there are only a few studies that consider graph visualization, whereas
most researchers focus on computation time. Thus, this study aims to construct
a novel picturing method called Half Butterfly Method (HBM) to address the
aforementioned scenario. Towards developing HBM, the concept of Wing
Strategy is introduced to create directions from one vertex to another vertex.
Then, these directions are used to map distinct vertices. In order to obtain the
distinct Hamiltonian circuits, the concept of matrix transpose is used to capture
the mirror image of that circuit. Several new theorems and lemmas are proved
in the decomposition of complete graphs into distinct Hamiltonian circuits.
Furthermore, the result of HBM is applied to list 𝑛! permutations and some
related theorems are established. In conclusion, this study successfully produced
a novel method to visualize distinct Hamiltonian circuit in complete graph
decomposition.
Keywords: Complete graph, Hamiltonian circuit, Graph decomposition
iii
Acknowledgement
In the name of Allah, the Most Gracious and the Most Merciful. Thank
you Allah for Your will and blessings that I am able to complete this thesis.
If I gave a name check to absolutely everyone who has helped make this
study possible, then these thank yous would run into tens of pages. The
generosity of spirit, inspiration and time which has been given by all those
involved is stunning, from all who have donated ideas and made sure that this
very interesting studies is piled high on the shelves. For those who have worked
around the clock to make this study every bit as good as the first, I thank you all.
Of course there are a few names who simply must get a mention. First
and foremost, special thanks to my supportive and enthusiastic supervisors,
Assoc. Prof. Dr. Haslinda binti Ibrahim and Dr. Sharmila binti Karim for their
guidance, patience and nurturing during the preparation of this thesis. Without
their mentoring and unbelievable amount of support that went beyond the call of
duty, this thesis would not have been possible.
I also want to express my special gratitude to Dr. Adyda binti Ibrahim
and all my friends for their tremendous helps, frequent advices and supports
along my full winding journey.
Finally, I am immensely grateful to my beloved mother, Che Seha
Ahmad for her endless supports, love and happiness that she have given me, as
well as for her tolerance of me during the frustrating times of my graduate
studies.
This thesis is dedicated to them.
Thank you.
iv
Declaration Associated with this Thesis
Darus, M. M., Ibrahim, H. & Karim, S. (2015). The construction of distinct
circuits of length six for complete graph 𝐾6 . International Journal of
Applied Mathematics and Statistics, 53(1), 17-31.
Darus, M. M., Ibrahim, H., Karim, S., Omar, Z., & Ahmad, N. (2014). Complete
graph 𝐾4 decomposition into circuits of length 4. Proceedings of the 21st
National Symposium on Mathematical Sciences (SKSM 21), 1605, 605610.
Darus, M. M., Ibrahim, H., & Karim, S. (2013). On the generation of distinct
circuits via complete graph decomposition. The First Innovation and
Analytics Conference and Exhibition, 26-32.
v
Table of Contents
Permission to Use ........................................................................................................... i
Abstrak .......................................................................................................................... ii
Abstract ........................................................................................................................ iii
Acknowledgement ........................................................................................................ iv
Declaration Associated with this Thesis.........................................................................v
Table of Contents ......................................................................................................... vi
List of Tables .............................................................................................................. viii
List of Figures .............................................................................................................. ix
List of Appendices........................................................................................................ xi
List of Symbols ........................................................................................................... xii
List of Abbreviations .................................................................................................. xiii
INTRODUCTION .............................................................. 1
1.1 General Reviews on Graph ........................................................................... 1
1.1.1 Hamiltonian Graph ............................................................................. 4
1.2 Graph Decomposition ................................................................................... 5
1.3 Motivation of the Study ................................................................................ 6
1.4 Research Questions ....................................................................................... 8
1.5 Research Objectives ...................................................................................... 8
1.6 Thesis Outline ............................................................................................... 8
RELATED MATERIALS AND DEFINITIONS ......... 10
2.1 The Shift-and-Rotate Strategy and the Butterfly Strategy .......................... 10
2.2 Related Materials and Definitions .............................................................. 11
SOME NUMERICAL EXAMPLES .......................... 17
3.1 Definitions and Terminologies ................................................................... 17
3.2 Some Examples of the Proposed Method ................................................... 18
3.2.1 Case 𝐾3 ............................................................................................. 19
3.2.2 Case 𝐾4 ............................................................................................. 21
3.2.3 Case 𝐾5 ............................................................................................. 31
3.2.4 Case 𝐾6 ............................................................................................. 42
vi
THE HALF BUTTERFLY METHOD ........................ 57
4.1 Wing Strategy ............................................................................................. 59
4.2 𝐾𝑛 Decomposition into Distinct Hamiltonian Circuits with Different Path 72
4.3 Conceptual Results on the HBM................................................................. 79
4.4 Conclusion .................................................................................................. 89
AN APPLICATION OF THE HALF BUTTERFLY
METHOD ........................................................................................................ 90
5.1 The Permutation of 𝑛 Elements .................................................................. 90
5.2 The Permutation of Three Elements using the HBM.................................. 91
5.3 The Permutation of Four Elements using the HBM ................................... 92
5.4 The Permutation of Five Elements using the HBM .................................... 94
5.5 The Permutation of 𝑛 Elements using the HBM ........................................ 98
CONCLUSION ................................................................. 102
6.1 Summary of Each Chapter ........................................................................ 102
6.2 Contribution of the Study.......................................................................... 104
6.3 Suggestion for Future Research ................................................................ 105
6.4 Discussion ................................................................................................. 112
REFERENCES .............................................................................................. 113
APPENDICES ............................................................................................... 117
vii
List of Tables
Table 4.1: Differences between BS and HBM. ...................................................................... 57
Table 4.2: The steps of HBM. ................................................................................................ 58
Table 4.3: Creating directions in block 1. .............................................................................. 68
Table 4.4: Creating directions in block 2. .............................................................................. 69
Table 4.5: Creating directions in block 3. .............................................................................. 70
Table 4.6: Creating directions in the final block. ................................................................... 71
Table 5.1: The permutations of three elements ...................................................................... 92
Table 5.2: The permutations of four elements. ...................................................................... 93
Table 5.3: The permutations of five elements........................................................................ 94
Table 6.1: The number of different structure of distinct Hamiltonian circuits from 𝐾𝑛 . ..... 105
Table 6.2: Analysis of distinct Hamiltonian circuits with different structure for 𝐾5 . .......... 108
Table 6.3: Analysis of distinct Hamiltonian circuits with different structures for 𝐾6 . ........ 110
viii
List of Figures
Figure 1.1. Routes for five historical sites in Malacca (this map has been modified from
Google maps) ........................................................................................................................... 1
Figure 2.1. Isomorphic graphs ............................................................................................... 14
Figure 3.1. Circuit 𝐴 is isomorphic to circuit 𝐵 ..................................................................... 18
Figure 3.2. A complete graph 𝐾3 ........................................................................................... 19
Figure 3.3. The circuits from 𝐾3 ............................................................................................ 19
Figure 3.4. The distinct Hamiltonian circuit with different paths from 𝐾3 ............................ 20
Figure 3.5. Several circuits from 𝐾4 ...................................................................................... 21
Figure 3.6. The same circuits from 𝐵 with different starting point........................................ 22
Figure 3.7. The same circuits from 𝐶 with different starting point ........................................ 22
Figure 3.8. The same circuits from 𝐹 with different starting point ........................................ 23
Figure 3.9. Three distinct Hamiltonian circuits with different paths from 𝐾4 ....................... 24
Figure 3.10. Creating directions in block 1 for 𝐾4 ................................................................ 25
Figure 3.11. Creating directions in block 2 for 𝐾4 ................................................................ 26
Figure 3.12. Creating directions in block 3 for 𝐾4 ................................................................ 27
Figure 3.13. Fix-and-shift for 𝐾4 where vertex 1 is fixed ..................................................... 28
Figure 3.14. Fix-and-shift for 𝐾4 where vertex 2 is fixed ..................................................... 28
Figure 3.15. Fix-and-shift for 𝐾4 where vertex 3 is fixed ..................................................... 29
Figure 3.16. Fix-and-shift for 𝐾4 where vertex 4 is fixed ..................................................... 29
Figure 3.17. The three distinct Hamiltonian circuits with different paths from 𝐾4 ............... 31
Figure 3.18. A complete graph 𝐾5 ......................................................................................... 32
Figure 3.19. Creating directions in block 1 for 𝐾5 ................................................................ 33
Figure 3.20. Creating directions in block 2 for 𝐾5 ................................................................ 34
Figure 3.21. Creating directions in block 3 for 𝐾5 ................................................................ 35
Figure 3.22. Creating directions in block 4 for 𝐾5 ................................................................ 36
Figure 3.23. Fix-and-shift for 𝐾5 where vertex 1 is fixed ..................................................... 37
Figure 3.24. Fix-and-shift for 𝐾5 where vertex 2 is fixed ..................................................... 38
Figure 3.25. Fix-and-shift for 𝐾5 where vertex 3 is fixed ..................................................... 38
Figure 3.26. Fix-and-shift for 𝐾5 where vertex 4 is fixed ..................................................... 39
Figure 3.27. Fix-and-shift for 𝐾5 where vertex 5 is fixed ..................................................... 39
ix
Figure 3.28. Drawing the circuits for 𝐾5 ............................................................................... 41
Figure 3.29. Several circuits from 𝐾6 .................................................................................... 43
Figure 3.30. Creating directions in block 1 for 𝐾6 ................................................................ 45
Figure 3.31. Creating directions in block 2 for 𝐾6 ................................................................ 46
Figure 3.32. Creating directions in block 3 for 𝐾6 ................................................................ 47
Figure 3.33. Creating directions in block 4 for 𝐾6 ................................................................ 48
Figure 3.34. Creating directions in block 5 for 𝐾6 ................................................................ 49
Figure 3.35. Fix-and-shift for 𝐾6 where vertex 1 is fixed ..................................................... 50
Figure 3.36. Fix-and-shift for 𝐾6 where vertex 2 is fixed ..................................................... 51
Figure 3.37. Fix-and-shift for 𝐾6 where vertex 3 is fixed ..................................................... 52
Figure 3.38. Fix-and-shift for 𝐾6 where vertex 4 is fixed ..................................................... 52
Figure 3.39. Fix-and-shift for 𝐾6 where vertex 5 is fixed. .................................................... 53
Figure 3.40. Fix-and-shift for 𝐾6 where vertex 6 is fixed ..................................................... 53
Figure 3.41. Drawing the circuits for 𝐾6 ............................................................................... 55
Figure 4.1. Creating direction using WS................................................................................ 59
Figure 4.2. Creating directions in block 1 for 𝐾𝑛 .................................................................. 73
Figure 4.3. Step of fix-and-shift, vertex 𝑥1 is fixed ............................................................... 74
Figure 4.4. Step of fix-and-shift, vertex 𝑥2 is fixed ............................................................... 75
Figure 4.5. Step of fix-and-shift, vertex 𝑥𝑛 is fixed ............................................................... 76
Figure 4.6. Drawing the circuits for all 𝐾𝑛 ............................................................................ 78
Figure 5.1. Hamiltonian circuit with 𝑛 vertices. .................................................................... 98
Figure 6.1. Two different structures of distinct Hamiltonian circuits with different paths
from 𝐾4 ................................................................................................................................ 107
Figure 6.2. Four different structures of distinct Hamiltonian circuits with different paths
from 𝐾5 ................................................................................................................................ 107
Figure 6.3. Fourteen different structures of distinct Hamiltonian circuits with different paths
from 𝐾6 . ............................................................................................................................... 109
x
List of Appendices
Appendix A Fix-and-shift for 𝐾4 ......................................................................................... 117
Appendix B Fix-and-shift for𝐾5 .......................................................................................... 118
Appendix C Twelve distinct mappings for 𝐾5 ..................................................................... 124
Appendix D Twelve distinct HC with different path from 𝐾5 ..........................................125
Appendix E Fix-and-shift for 𝐾6 ......................................................................................... 127
Appendix F Sixty distinct mappings for 𝐾6 ......................................................................... 137
Appendix G Sixty distinct HC with different path from 𝐾6 ................................................ 140
Appendix H Creating direction in block 1 for 𝐾𝑛 ................................................................ 150
Appendix I Creating direction in block 2 for 𝐾𝑛 ................................................................. 152
Appendix J Creating direction in block 3 for 𝐾𝑛 ................................................................. 154
Appendix K Creating direction in final block for 𝐾𝑛 .......................................................... 156
Appendix L Sequences of direction for 𝐾𝑛 .......................................................................... 158
xi
List of Symbols
𝑮
A graph consists of vertices and edges
𝑽(𝑮)
The vertices of a graph 𝐺
𝑬(𝑮)
The edges of a graph 𝐺
𝑮(𝑽, 𝑬)
A graph 𝐺 with the set of vertices 𝑉(𝐺) and edges 𝐸(𝐺)
𝑲𝒏
A complete graph with 𝑛 vertices
𝑪𝒏
A cycle with 𝑛 vertices
𝑾𝒏
A complete directed graph with 𝑛 vertices
∗
A circuit with 𝑛 vertices
𝑲𝒏
𝑪𝒏
A wheel with 𝑛 vertices
∗
𝑷𝒏
A path with 𝑛 vertices
𝑲𝒎,𝒏
A bipartite graph with two disjoint sets, 𝑚 and 𝑛
𝑲𝒄𝒎,𝒏
The complement set of bipartite graph 𝐾𝑚,𝑛
𝑪𝑻𝒏
A complete-transposition graph with 𝑛 vertices
𝑺𝒌
A star graph
𝑫𝒓×𝒄
A Cartesian product with 𝑟 rows and 𝑐 columns
deg (𝒖)
The degree of vertex 𝑢 where 𝑢 is a vertex in any graph
𝒙𝒊
A vertex in any graph, where 1 ≤ 𝑖 ≤ 𝑛
𝒏 ∈ ℤ+
The total number of vertices of any graph, where 𝑛 is positive
𝝀
The block for each direction in WS
𝜸
Total permutations of 𝑛 elements
xii
List of Abbreviations
WS
Wing Strategy
BS
Butterfly Strategy
S-R
Shift-and-Rotate Strategy
HBM
Half Butterfly Method
xiii
INTRODUCTION
Graph decomposition is an important research in graph theory because it can
model several networks in our daily life such as social network, railway network
and internet network (Kante, 2008). In the literature, several studies regarding
graph decomposition have been focused such as the studies done by Granville,
Moisiadis and Rees (1989); Adams, Bryant, Forbes and Griggs, (2012); and Yuan
and Kuang, (2012); among others.
1.1 General Reviews on Graph
Let say, we are planning to visit five historic sites in Malacca without visiting
each site more than once. Figure 1.1 shows the routes connecting each site. From
the map, we need to manage possible route which will make sure we will never
visit any site twice.
Jonker
Walk
Cheng
Ho Art
Museum
Car park
Maritime
Museum
River
Cruise
Taming Sari
Tower
Figure 1.1. Routes for five historical sites in Malacca (this map has been
modified from Google maps)
1
The contents of
the thesis is for
internal user
only
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