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**Hamilton Graph Theory โดย 1. นายธนพัฒน์ อัตถกิจมงคล ม. 6/7 เลขที่ 14 2**

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กราฟแฮมิลตัน เซอร์วิลเลียม โรแวน แฮมิลตัน (Sir William Rowan Hamilton) นักคณิตศาสตร์ชาวไอริช ได้ประดิษฐ์ก้อนไม้ที่มี 20 มุม และประกอบด้วยรูปห้าเหลี่ยมด้านเท่าจำนวน 12 หน้า

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กราฟแฮมิลตัน ปัญหาก็คือการหาเส้นทางโดยเริ่มจากเมืองหนึ่ง ๆ แล้วไล่ตามขอบของก้อนไม้ไปเรื่อย ๆ เพื่อที่จะแวะผ่านทุก เมือง เมืองละ 1 ครั้ง แล้ววนกลับสู่เมืองที่ตั้งต้น

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วิถีและวงจรแฮมิลตัน ให้ G = (V,E) เป็นกราฟ วิถีแฮมิลตัน (Hamiltonian Path) คือวิถีในกราฟซึ่งใช้จุดทุกจุดเพียงจุดละ 1 ครั้ง โดยไม่จำเป็นต้องใช้เส้นเชื่อมครบทุกเส้น และถ้าจุดยอดเริ่มต้นกับจุดยอดสุดท้ายของวิถีเป็นจุดเดียวกัน จะเรียกว่า วงจรแฮมิลตัน (Hamiltonian Circuit)

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**ตัวอย่าง การหาวัฏจักรแฮมิลโทเนียนของกราฟ**

กราฟมี วัฏจักรแฮมิลตัน โดยมีเส้นทางเดิน คือ a , b , c , h , g , e , f , d , a a b c h g e f d . G

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

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24-Cell The 24-cell is a finite regular four-dimensional. It is also known as the hyperdiamond or icositetrachoron, and is composed of 24 octahedra, 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.

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

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

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

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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.

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

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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.

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Brinkmann Graph The Brinkmann graph is a weakly regular quartic graph on 21 vertices and 42 edges.

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

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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.

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Cuboctahedral Graph An Archimedean symmetric 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.

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

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

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**Disdyakis Dodecahedral Graph**

The disdyakis dodecahedral graph is Archimedean dual graph which is the skeleton of the disdyakis dodecahedron.

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**Disdyakis Dodecahedron**

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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.

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Dyck Graph The unique cubic symmetric graph on 32 nodes 48 edges. It is nonplanar.

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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.

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

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

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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.

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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).

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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).

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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

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**Truncated Tetrahedral Graph**

The truncated tetrahedral graph is the cubic Archimedean graph on 12 nodes and 18 edges that is the skeleton of the truncated tetrahedron.

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**Truncated Octahedral Graph**

The truncated octahedron graph is the cubic Archimedean graph on 24 nodes and 36 edges that is the skeleton of the truncated octahedron.

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**Truncated Icosahedral Graph**

The truncated icosahedral graph is the cubic Archimedean graph on 60 nodes and 90 edges that is the skeleton of the truncated icosahedron. A number of embeddings are shown above.

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**Truncated Dodecahedral Graph**

The cubic Archimedean graph on 60 nodes and 90 edges that is the skeleton of the truncated dodecahedron.

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**Truncated Cubical Graph**

The cubic Archimedean graph on 24 nodes and 36 edges that is the skeleton of the truncated cube.

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

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**Triakis Tetrahedral Graph**

The triakis tetrahedral graph is Archimedean dual graph which is the skeleton of the triakis tetrahedron.

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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 .

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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.

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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}.

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Square Graph The cycle graph C4 .

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**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.

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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.

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**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.

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**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.

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**Robertson-Wegner Graph**

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

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

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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.

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**Pentakis Dodecahedral Graph**

The pentakis dodecahedral graph is Archimedean dual graph which is the skeleton of the disdyakis triacontahedron.

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**Pentagonal Icositetrahedral Graph**

The pentagonal icositetrahedral graph is the Archimedean dual graph which is the skeleton of the pentagonal icositetrahedron.

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**Pentagonal Hexecontahedral Graph**

The pentagonal hexecontahedral graph is the Archimedean dual graph which is the skeleton of the pentagonal hexecontahedron.

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Pappus Graph A cubic symmetric distance-regular graph on 18 vertices, illustrated above in three embeddings.

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Octahedral Graph The 6-node 12-edge Platonic graph having the connectivity of the octahedron

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Möbius-Kantor Graph The unique cubic symmetric graph on 16 nodes, illustrated above in two embeddings.

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Meringer Graph The Meringer graph has 30 nodes. It has 75 edges, girth 5, diameter 3, chromatic number 3, and is a quintic graph. The order of its automorphism group is 96.

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McGee Graph The McGee graph was discovered by McGee (1960) and proven unique by Tutte (1966; Wong 1982). It has 24 nodes, 36 edges, girth 7, diameter 4, and is a cubic graph.

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Levi Graph "The" Levi graph (right figure) is a graph based on Desargues' configuration which consists of the union of the two leftmost subgraphs illustrated above. It has 30 nodes and 45 edges. It has girth 8, diameter 4, chromatic number 2, and automorphism group order 1440.

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Kittell Graph The Kittell graph is a planar graph on 23 nodes and 63 edges that tangles the Kempe chains in Kempe's algorithm and thus provides an example of how Kempe's supposed proof of the four-color theorem fails.

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**Icosidodecahedral Graph**

A symmetric quartic graph on 30 nodes and 60 edges corresponding to the skeleton of the icosidodecahedron. It has graph diameter 5, graph radius 5, and chromatic number 3. It is also Hamiltonian.

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Icosahedral Graph The icosahedral graph is the Platonic graph whose nodes have the connectivity of the icosahedron, illustrated above in a number of embeddings. The icosahedral graph has 12 vertices and 30 edges.

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House Graph The house graph is a simple graph on 5 nodes and 6 edges whose name derives from its resemblance to a schematic illustration of a house with a roof.

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**Hoffman-Singleton Graph**

The Hoffman-Singleton graph is the graph on 50 nodes and 175 edges that is the only regular graph of vertex degree 7, diameter 2, and girth 5.

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Hoffman Graph The Hoffman graph is the bipartite graph on 16 nodes and 32 edges.

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Heawood Graph The Heawood graph is the cage graph illustrated above in a number of embeddings. It is 4-transitive, but not 5-transitive (Harary 1994, p. 173).

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Harries-Wong Graph The Harries-Wong graph is one of the three (10,3) -cage graphs, the other two being the (10,3) -cage Balaban graph and the Harries graph.

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Harries Graph The Harries graph is one of the three (10,3) -cage graphs, the other two being the (10,3) -cage Balaban graph and the Harries-Wong graph.

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Harborth Graph The Harborth graph is the smallest known 4-regular matchstick graph. It is therefore both planar and unit-distance. It has 104 edges and 52 vertices. This graph was named after its discoverer H. Harborth, who first presented it to a general public in 1986 (Harborth 1986, Petersen 1996, Gerbracht 2006).

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Grünbaum Graph The Grünbaum graph can be constructed from the dodecahedral graph by adding an additional ring of five vertices around the perimeter and cyclically connecting each new vertex to three others as shown above (left figure). A more symmetrical embedding is shown in the right figure above. The Grünbaum graph has 25 vertices and 50 edges.

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Grötzsch Graph The Grötzsch graph is smallest triangle-free graph with chromatic number four. It is identical to the Mycielski Graph of order four. It has 11 vertices and 20 edges. It is Hamiltonian, but nonplanar.

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**Great Rhombicuboctahedral Graph**

The cubic Archimedean graph on 48 nodes and 72 edges that is the skeleton of the great rhombicuboctahedron.

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**Great Rhombicosidodecahedral Graph**

The great rhombicosidodecahedral graph is the Archimedean graph on 120 vertices and 180 edges that is the skeleton of the great rhombicosidodecahedron. It is cubic, has chromatic number 2, and is planar and Hamiltonian.

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Gray Graph The Gray graph is a semisymmetric cubic graph on 54 vertices. It was discovered by Marion C. Gray in 1932, and was first published by Bouwer (1968). Malnic et al. (2004) showed that the Gray graph is indeed the smallest possible semisymmetric cubic graph.

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Gewirtz Graph The Gewirtz graph, sometimes also called the Sims-Gewirtz graph (Brouwer), is an integral graph on 56 nodes and 280 edges that is also a regular graph of order 10.

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Frucht Graph The Frucht graph is smallest cubic identity graph (Skiena 1990, p. 185). It has 12 vertices and 18 edges. It is also both planar and Hamiltonian.

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**ทฤษฎีบทของดิแรก(Dirac’s theoem)**

กราฟเชิงเดียวที่มีจุดยอด n จุด เมื่อ n >= 3 และจุดยอดทุกจุด มีดีกรีอย่างน้อย n/2 แล้วกราฟดังกล่าวเป็นกราฟแฮมิลตัน

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