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New Methods for Magic Total Labelings of Graphs

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New Methods for Magic Total Labelings of Graphs NEW METHODS FOR MAGIC TOTAL LABELINGS OF GRAPHS A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Inne Singgih IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE Dalibor Froncek, Sylwia Cichacz April, 2015 c Inne Singgih 2015 ALL RIGHTS RESERVED Acknowledgements There are many people that have earned my gratitude for their contribution to my time in graduate school. I would like to express my sincere gratitude to my advisor, Dalibor Froncek for encouraging me to apply to UMD and all his support over the past two years. I am especially grateful for suggesting a topic and all the guidance he gave me during writing this thesis despite of his busy schedule. In addition, I would like to thank my co-advisor, Sylwia Cichacz, for her inputs and discussion for this paper over the past year. I would also like to thank Joseph Gallian and Barry James for serving on my committee. Last but not least, I would like to thank fellow students in the Department of Mathe- matics and Statistics for their helpful discussions and the warmth they shared during my stay in the cold Duluth winter. i Abstract A vertex magic total (VMT) labeling of a graph G = (V; E) is a bijection from the set of vertices and edges to the set of numbers defined by λ : V [ E ! f1; 2;:::; jV j + jEjg P so that for every x 2 V and some integer k, w(x) = λ(x) + y:xy2E λ(xy) = k. An edge magic total (EMT) labeling is a bijection from the set of vertices and edges to the set of numbers defined by λ : V [ E ! f1; 2;:::; jV j + jEjg so that for every xy 2 E and some integer k, w(xy) = λ(x) + λ(y) + λ(xy) = k. Numerous results on labelings of many families of graphs have been published. In this thesis, we include methods that expand known VMT/EMT labelings into VMT/EMT labelings of some new families of graphs, such as unions of cycles, unions of paths, cycles with chords, tadpole graphs, braid graphs, triangular belts, wheels, fans, friendships, and more. ii Contents Acknowledgements i Abstract ii List of Tables vi List of Figures ix 1 Introduction 1 1.1 Basic Graph Terminology . 1 1.2 Unions, Products and Joins of Graphs . 8 1.3 Graph Labeling . 11 1.3.1 Rosa's Valuation and Graceful Labeling . 11 1.4 Magic Labeling . 12 1.5 Permutation Cycles and Kotzig Arrays . 13 1.6 Other Definitions . 14 2 Related Results 16 2.1 Cycles . 17 2.2 Unions of Identical Cycles mCn ....................... 17 2.3 Unions of Two Cycles of Distinct Length Cn1 [ Cn2 . 18 2.4 Unions of Cycles Cn [ hC3 .......................... 19 iii 2.5 Unions of Cycles and Paths Cn1 [ Pn2 ................... 19 t 2.6 A Cycle with a Chord Cn .......................... 20 2.7 Lexicographic Product Cn ◦ K2 ....................... 20 2.8 Cartesian Product Pm Cn ......................... 21 2.9 Cartesian Product P2 Pn ......................... 21 2.10 Unions of Paths . 22 2.11 (n; t)-tadpole graph . 22 2.12 Friendship graph F rn, Fan fn, Wheel Wn . 23 2.13 P2n(+)Nm and (P2 [ mK1) + N2 ...................... 23 2.14 Braid B(n), Triangular Belt TB(α), Umbrella U(m; n) . 24 2.15 Jellyfish J(m; n), Incomplete Mongolian Ger MT (m; h) . 24 3 Results on Vertex and Super Vertex Magic Total Labelings 25 3.1 Method 1: Extending the Length of Cycles . 25 3.2 Method 2: Multiplying the Number of Cycles . 28 3.3 Unions of an Even Number of Identical Even Cycles . 29 3.4 Unions of Non-Identical Cycles m Cn1(2r+1) [ Cn2(2r+1) . 34 3.5 Unions of Cycles m Cn(2r+1) [ hC3 .................... 39 3.6 Further Research on VMT/SVMT Labelings . 40 4 Results on Edge and Super Edge Magic Total Labelings 43 4.1 Method 4: Extending the Length of Cycles, Multiplying the Number of Paths . 43 4.2 Unions of Cycles and Paths m Cn1(2r+1) [ (2r + 1)Pn2 . 44 [c]t 4.3 Cycle with c Chords Cn .......................... 50 4.4 Lexicographic Product Cn ◦ K2 ....................... 55 4.5 Cartesian Product P2 Cn ......................... 61 4.6 Cartesian Product P2 Pn ......................... 63 4.7 Unions of Paths mPn ............................. 65 iv 4.8 (n; t)-tadpole and Mutated (m; n; t)-tadpole . 68 4.9 Method 4 Performed on Friendship Graph F rn . 75 4.10 Method 4 Performed on Fan fn ....................... 77 4.11 Method 4 Performed on Wheel Wn ..................... 82 4.12 Method 4 Performed on Umbrella U(m; n) . 84 4.13 P2n(+)Nm ! C(2n+1)(2r+1)[+]Nm ...................... 88 4.14 Method 4 Performed on (P2 [ mK1) + N2 . 93 4.15 Unions of Braids mB(n)........................... 96 4.16 Unions of Triangular Belts mT B(α) .................... 97 4.17 Jellyfish J(m; n) ! Rambutan R(t; m; n) . 99 4.18 Incomplete Mongolian Ger MT (m; h) to a Circus Tent . 102 References 104 Appendix A. Summary of Known VMT and SVMT Labeling 107 Appendix B. Summary of Known EMT and SEMT Labeling 110 Appendix C. Summary of New Results 115 v List of Tables 3.1 Tables for 2C6 ! 2C18 ............................ 30 3.2 Tables for C3 [ C6 ! C9 [ C18 ....................... 37 3.3 Tables for 2C6 ! 4C12 [ 2C6 ........................ 41 4.1 Tables for C4 [ CP2 ! C12 [ 3P2 ...................... 46 4.2 Tables for C4 [ P2 ! C20 [ 5P2 ....................... 47 2 [3]9 4.3 Table for C7 ! C21 (chord) . 50 2 [3]9 4.4 Table for C7 ! C21 (cycle) . 51 2 [5]9 4.5 Table for C7 ! C35 (cycle) . 52 2 [5]9 4.6 Table for C7 ! C35 (chord) . 52 2 [3]5 4.7 Table for C7 ! C21 (chord) . 53 2 [3]5 4.8 Table for C7 ! C21 (cycle) . 54 4.9 Tables for Decomposed ! Extended decomposed C4 ◦ K2 . 56 4.10 Tables for Decomposed ! Extended decomposed C4 ◦ K2 . 57 4.11 Tables for P2 C3 ! P2 C9 (vertical paths) . 62 4.12 Tables for P2 C3 ! P2 C9 (cycles) . 62 4.13 Tables for P2 P5 ! m (P2 P5) for P5 . 63 4.14 Tables for P2 P5 ! m (P2 P5) for P2 . 64 4.15 Tables for P4 [ P2 ! 3(P4 [ P2)....................... 65 4.16 Tables for P3 [ 3P2 ! 5(P3 [ 3P2) ..................... 66 4.17 Tables for 2P4 ! 6P4 ............................. 67 vi 4.18 Table for (4; 2)-tadpole ! mutated (3; 12; 2)-tadpole (body) . 73 4.19 Table for (4; 2)-tadpole ! mutated (3; 12; 2)-tadpole (tail) . 73 4.20 Tables for Method 4 performed on F r3 with factor (2r + 1) = 3 . 76 4.21 Table for Method 4 performed on f4 with factor (2r + 1) = 3 (path) . 77 4.22 Tables for Method 4 performed on f4 with factor (2r + 1) = 3 (cycles) . 78 4.23 Table for Method 4 performed on f4 with factor (2r + 1) = 5 (paths) . 78 4.24 Tables for Method 4 performed on f4 with factor (2r + 1) = 5 (cycles) . 79 4.25 Tables for Method 4 performed on f6 with factor (2r + 1) = 5 (cycles) . 80 4.26 Tables for Method 4 performed on f6 with factor (2r + 1) = 5 (paths) . 80 4.27 Tables for Method 4 performed on W6 with factor (2r + 1) = 3 (paths) . 82 4.28 Tables for Method 4 performed on W6 with factor (2r + 1) = 3 (cycles) 83 4.29 Tables for Method 4 performed on U(4; 3) with factor (2r + 1) = 3 (paths) 84 4.30 Tables for Method 4 performed on U(4; 3) with factor (2r + 1) = 3 (cycles) 85 4.31 Tables for Method 4 performed on U(6; 4) with factor (2r + 1) = 3 (cycles) 86 4.32 Tables for Method 4 performed on U(6; 4) with factor (2r + 1) = 3 (paths) 86 4.33 Table for Method 4 performed on U(6; 4) with factor (2r + 1) = 3 (stick) 87 4.34 Table for P2(+)N2 ! C15[+]5N1 (cycle) . 89 4.35 Table for P2(+)N2 ! C15[+]5N1 (path) . 89 4.36 Table for P2(+)N3 ! C9[+]3N2 (cycle) . 90 4.37 Tables for P2(+)N3 ! C9[+]3N2 (paths) . 90 4.38 Tables for P4(+)N2 ! C15[+]3N1 ...................... 92 4.39 Tables for (P2 [ 3K1) + N2 ! C12[+]N3 with 3 chords (for P2) . 94 4.40 Tables for (P2 [ 3K1) + N2 ! C12[+]N3 with 3 chords (for C4) . 94 4.41 Tables for (P2 [ 3K1) + N2 ! C12[+]N3 with 3 chords (for 3P3) . 94 4.42 Tables for B(3) ! 3B(3) . 96 4.43 Tables for TB #3 ! 3TB #3 (horizontal paths) . 98 4.44 Tables for TB #3 ! 3TB #3 (vertical paths) . 98 4.45 Tables for TB #3 ! 3TB #3 (diagonal paths) . 99 vii 4.46 Tables for J(2; 3) ! R(3; 3; 2) (body) . 100 4.47 Tables for J(2; 3) ! R(3; 3; 2) (for m = 2 tentacles) . 100 4.48 Tables for J(2; 3) ! R(3; 3; 2) (for n = 3 tentacles) . 101 4.49 Tables for MT (3; 2) ! CT (9; 2; 3) (roof) . 102 4.50 Tables for MT (3; 2) ! CT (9; 2; 3) (vertical paths) . 103 4.51 Tables for MT (3; 2) ! CT (9; 2; 3) (cycles) . 103 A.1 Summary of Vertex-magic Total Labelings . 107 A.2 Summary of Vertex-magic Total Labelings (cont.) . 108 A.3 Summary of Super Vertex-magic Total Labelings . 109 B.1 Summary of Edge-magic Total Labelings . 110 B.2 Summary of Edge-magic Total Labelings (cont.) .
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