Hamilton Graph Theory โดย 1. นายธนพัฒน์ อัตถกิจมงคล ม.6/7 เลขที่ 14 2. นายเศรษฐพงศ์ อัศวรัตน์ ม.6/7 เลขที่ 20 3. นายสุภา

1. Hamilton Graph Theoryโดย1. นายธนพัฒน์ อัตถกิจมงคล ม.6/7 เลขที่ 142. นายเศรษฐพงศ์ อัศวรัตน์ ม.6/7 เลขที่ 203. นายสุภาเทพ ตัณศิริชัยยา ม.6/7 เลขที่ 22

2. กราฟแฮมิลตัน เซอร์วิลเลียมโรแวนแฮมิลตัน (Sir William Rowan Hamilton) นักคณิตศาสตร์ชาวไอริชได้ประดิษฐ์ก้อนไม้ที่มี 20 มุมและประกอบด้วยรูปห้าเหลี่ยมด้านเท่าจำนวน 12 หน้า

3. กราฟแฮมิลตัน ปัญหาก็คือการหาเส้นทางโดยเริ่มจากเมืองหนึ่งๆแล้วไล่ตามขอบของก้อนไม้ไปเรื่อยๆเพื่อที่จะแวะผ่านทุกเมืองเมืองละ 1 ครั้งแล้ววนกลับสู่เมืองที่ตั้งต้น

4. วิถีและวงจรแฮมิลตัน ให้ G = (V,E) เป็นกราฟวิถีแฮมิลตัน (Hamiltonian Path) คือวิถีในกราฟซึ่งใช้จุดทุกจุดเพียงจุดละ 1 ครั้งโดยไม่จำเป็นต้องใช้เส้นเชื่อมครบทุกเส้นและถ้าจุดยอดเริ่มต้นกับจุดยอดสุดท้ายของวิถีเป็นจุดเดียวกันจะเรียกว่าวงจรแฮมิลตัน (Hamiltonian Circuit)

5. b a c h d . e g f ตัวอย่าง การหาวัฏจักรแฮมิลโทเนียนของกราฟ กราฟมีวัฏจักรแฮมิลตันโดยมีเส้นทางเดิน คือ a , b , c , h , g , e , f , d , a G

6. 16-Cell • The 16-cell is the finite regular four-dimensional.It is also known as the hyperoctahedron or hexadecachoron, and its composed of 16tetrahedra, with 4 to an edge. It has 8 vertices, 24 edges, and 32 faces. The 16-cell. It has distinct nets.

7. 24-Cell • The 24-cell is a finite regular four-dimensional. It is also known as the hyperdiamond or icositetrachoron, and is composed of 24octahedra, with 3 to an edge. The 24-cell has 24 vertices and 96 edges. It is one of the six regular polychora. The 24-cell has distinct nets.

8. 120-Cell • The 120-cell is a finite regular four-dimensional, also known as the hyperdodecahedron or hecatonicosachoron, and composed of 120dodecahedra, with 3 to an edge, and 720pentagons.The 120-cell has 600 vertices and 1200 edges.

10. 600-Cell • The 600-cell is the finite regular four-dimensional. It is also known as the hypericosahedron or hexacosichoron. It is composed of 600tetrahedra, with 5 to an edge. The 600-cell has 120 vertices and 720 edges.

11. Balaban 10-Cage • It is a Hamiltonian graph and has 91440 Hamiltonian cycles. The Balaban 10-cage is one of the three -cage graphs.

12. Bidiakis Cube • The 12-vertex graph consisting of a cube in which two opposite faces (say, top and bottom) have edges drawn across them which connect the centers of opposite sides of the faces in such a way that the orientation of the edges added on top and bottom are perpendicular to each other.

13. Biggs-Smith graph • The Biggs-Smith graph is cubic symmetric graph on 102 vertices and 153 edges that is also distance-regular.

14. Bislit Cube • The bislit cube is the 8-vertex simple graph consisting of a cube in which two opposite faces have polyhedron diagonals oriented perpendicular to each other.

15. Brinkmann Graph • The Brinkmann graph is a weakly regularquartic graph on 21 vertices and 42 edges.

16. Clebsch Graph • The Clebsch graph, also known as the Greenwood-Gleason graph , is a strongly regularquintic graph on 16 vertices and 40 edges

17. Cubical Graph • The cubical graph is the Platonic graph corresponding to the connectivity of the cube. It is equivalent to the generalized Petersen graph. The cubical graph has 8 nodes, 12 edges.

18. Cuboctahedral Graph • An Archimedeansymmetric quartic graph on 12 nodes and 24 edges that is the skeleton of the cuboctahedron. The cuboctahedral graph is the line graph of the cubical graph.

19. Desargues Graph • The Desargues graph is a cubic symmetric graphdistance-regular graph on 20 vertices and 30 edges.

20. Diamond Graph • The diamond graph is the simple graph on 4 nodes and 5 edges illustrated above.

21. Disdyakis Dodecahedral Graph • The disdyakis dodecahedral graph is Archimedean dual graph which is the skeleton of the disdyakis dodecahedron.

22. Disdyakis Dodecahedron

23. Dodecahedral Graph • The dodecahedral graph is the Platonic graph corresponding to the connectivity of the vertices of a dodecahedron. The dodecahedral graph has 20 nodes, 30 edges.

24. Dyck Graph • The unique cubic symmetric graph on 32 nodes48 edges. It is nonplanar.

25. Errera Graph • The Errera graph is the 17-node planar graph. It is an example of how Kempe's supposed proof of the four-color theorem fails.

26. Folkman Graph • The Folkman graph is a semisymmetric graph that has the minimum possible number of nodes (20)

27. Foster Graph • The" Foster graph is the cubic symmetric graph on 90 vertices that has 135 edges

28. Wong Graph The Wong graph is one of the four (5,5) -cage graphs. Like the other (5,5) -cages, the Wong graph has 30 nodes. It has 75 edges, girth 5, diameter 3, chromatic number 4, and is a quintic graph.

29. Wells Graph The Wells graph is a quintic graph on 32 nodes and 80 edges that is the unique distance-regular graph with intersection array (5, 4, 1, 1; 1, 1, 4, 5) . It is a double cover of the complement of the Clebsch graph (Brouwer et al. 1989, p. 266).

30. Utility Graph The utility problem posits three houses and three utility companies--say, gas, electric, and water--and asks if each utility can be connected to each house without having any of the gas/water/electric lines/pipes pass over any other. This is equivalent to the equation "Can a planar graph be constructed from each of three nodes ('houses') to each of three other nodes ('utilities')?" This problem was first posed in this form by H. E. Dudeney in 1917 (Gardner 1984, p. 92).

31. Unitransitive Graph A graph G is n -unitransitive if it is connected, cubic, n-transitive, and if for any two n-routes W1 and W2

32. Truncated Tetrahedral Graph The truncated tetrahedral graph is the cubicArchimedean graph on 12 nodes and 18 edges that is the skeleton of the truncated tetrahedron.

33. Truncated Octahedral Graph The truncated octahedron graph is the cubicArchimedean graph on 24 nodes and 36 edges that is the skeleton of the truncated octahedron.

34. Truncated Icosahedral Graph The truncated icosahedral graph is the cubicArchimedean graph on 60 nodes and 90 edges that is the skeleton of the truncated icosahedron. A number of embeddings are shown above.

36. Truncated Dodecahedral Graph The cubicArchimedean graph on 60 nodes and 90 edges that is the skeleton of the truncated dodecahedron.

37. Truncated Cubical Graph The cubicArchimedean graph on 24 nodes and 36 edges that is the skeleton of the truncated cube.

38. Triangle Graph The triangle graph is the cycle graph C3 , which is also the complete graph K3 .

39. Triakis Tetrahedral Graph The triakis tetrahedral graph is Archimedean dual graph which is the skeleton of the triakis tetrahedron.

40. Tetrahedral Graph The Platonic graph that is the unique polyhedral graph on four nodes which is also the complete graph K4 and therefore also the wheel graph W4 .

41. Tesseract The tesseract is composed of 8 cubes with 3 to an edge, and therefore has 16 vertices, 32 edges, 24 squares, and 8 cubes. It is one of the six regular polychora.

42. Sylvester Graph "The" Sylvester graph is a quintic graph on 36 nodes and 90 edges that is the unique distance-regular graph with intersection array {5, 4, 2; 1, 1, 4}.

43. Square Graph The cycle graph C4 .

44. Snub Dodecahedral Graph The snub dodecahedral graph is a quintic graph on 60 nodes and 150 edges that corresponds to the skeleton of the snub dodecahedron. The snub dodecahedral graph is planar and Hamiltonian, and has chromatic number 4. It is vertex-transitive, although not edge-transitive because some edges are part of three-circuits while others are not.

45. Snub Cubical Graph The snub cubical graph is the Archimedean graph on 24 nodes and 60 edges obtained by taking the skeleton of the snub cube. It is a quintic graph, is planar, Hamiltonian, and has chromatic number 3.

46. Small Rhombicuboctahedral Graph The small rhombicuboctahedral graph is a quartic graph on 24 nodes and 48 edges that corresponds to the skeleton of the small rhombicuboctahedron. It has graph diameter 5, graph radius 5, and chromatic number 3. It is also Hamiltonian. It is vertex-transitive, although not edge-transitive because some edges are part of three-circuits while others are not.

47. Small Rhombicosidodecahedral Graph The small rhombicosidodecahedral graph is a quartic graph on 60 nodes and 120 edges that corresponds to the skeleton of the small rhombicosidodecahedron. It has graph diameter 8, graph radius 8, and chromatic number 3. It is also Hamiltonian. It is vertex-transitive, although not edge-transitive because some edges are part of three-circuits while others are not.

48. Robertson-Wegner Graph The Robertson-Wegner graph has 30 nodes. It has 75 edges, girth 5, diameter 3, and chromatic number 4.

49. Robertson Graph The Robertson graph has 19 vertices and 38 edges.

50. Pentatope The pentatope is self-dual, has five three-dimensional facets (each the shape of a tetrahedron), 10 ridges (faces), 10 edges, and five vertices.