1 / 52

Graph Theory 1736Euler

dexter
Download Presentation

Graph Theory 1736Euler

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

    1. Graph Theory ????— ?1736?Euler????????? ??? ????????? 27 December 2003

    2. Problem of the Königsberg bridges ???????? ?Königsberg??????????????????????????????????? (??:?????????‚?????????????!) *Now Königsberg is Kaliningrad(Russia)!

    3. Königsberg

    4. ???????????????!

    5. Problem of the Königsberg bridges *????????”??????”! *Eueler ?????????????????????? 1736.

    6. Leonhard Euler Born: 15 April 1707 in Basel, Switzerland Died: 18 Sept 1783 in St Petersburg, Russia The advisor is Johann Bernoulli.

    7. Pictures of Euler

    8. ?(Graphs) ??G=(V,E)???????,????V(the vertex-set)????E(the edge-set)?????????,????V????

    10. ????(Degree) ?G???v?G ???? ??v??deg(v)????v??????????

    11. ??(Walk)???(Trail) ??G?,????(walk)?????W= (v1,e1,v2,e2,v3,…,vk-1,ek-1,vk)??v1,v2,…,vk ?G???‚e1,e2, …,ek-1?G???ei????vi?vi+1? ??G?,????(trail)?????W????????? ??G?,???????(closed trail)?????W??????????????????

    13. ??(Path)??(Cycle) ·??G?,????(path)?????W????????? ·??G??????????????,??G???(connected) ? ·?G???(cycle) C=(v1, v2,…,vk,vk+1) ??????v1, v2, …,vk?????v1=vk+1, ?????i?{1,2,…,k}, vivi+1??G???

    17. Euler??(Eulerian Trial)?Euler?(Eulerian circuit) ?G????Euler?(Eulerian circuit)???G?????????(Closed Trail)??G?????????G????Euler???Euler?? ?G????Euler??(Eulerian Trial)???G??????(Trail)??G?????

    20. Königsberg bridges

    21. ??(Examples)

    22. ??: ?G????Euler???????????(degree)?????

    23. How to prove it? It is easy to see that if G is eulerian then deg(v) is even for each v in V. Conversely, suppose G is a connected graph with deg(v) is even for each v in V. Then we can find a cycle C in G. Consider G-C, by induction on |E|, we can find that each connected part of G-C is eulerian. Combine C and eulerian subgraphs.

    25. Euler and Planar Graphs ??????Euler???????????

    26. Planar Graphs ?G?????(planar graph)??G??????????????? ?

    27. Euler’s Formula Euler’s Formula: v-e+f=2.

    29. History of Euler’s Formula Egypt: interested in regular polyhedra (4000 years ago). Greek: Archimedes (2000 years ago). Descartes(1640 A.D. founder of analytic geometry). Euler(1750 A.D.): A letter to Goldbach(1690~1764, German). A entirely different proof was given by Legendre in 1794.

    30. Archimedes(287 BC~212 BC, Sicily)

    31. Descartes(1696~1750, France)

    32. Legendre(1752~1833, France)

    33. ???(Trees) ?T????????T???????????????T??????

    34. ?? ???2????2?????????2??????1?(????????) 2. ?n??????n-1???(??????1.??) 3. ??G?n????n-1?????????????,?G???

    35. Two Trees

    36. ??: (Euler’s Formula) ??G????????v???, e ?,? f ??? v-e+f=2?

    37. 0

    38. 1

    39. 2

    40. 3

    41. 4

    42. 5

    43. The Proof of Euler’s Formula Suppose G is connected planar graph. If G contains a cycle, then G has at least two faces. Find an edge e which is between two distinct faces of G. G-e is till a connected planar graph and (the number of faces in G) = (the number of faces of G-e) +1. Repeated this way till we get a tree. Then calculate it!

    44. The Other Topics in Graph Theory

    45. ?????(Hamiltonian Graphs) Sir W. R. Hamilton ? 1856??????12?????????(????????????!)

    46. Sir William Rowan Hamilton (1805-1865, Ireland)

    47. ????(Graph Colorings) ???????? Frederick Guthrie (British) ?1852???? Frederick Guthrie ????????????????De Morgan? ??:(Guthrie) ??????????? ?1976???? Appel and Haken ????????????????? (????????? 1400???)

    49. DeMorgan(1806~1871, England)

    50. Applications ???? ???? ????

    51. ????????????? ??:??????????????????????,???????????,????????????????????,?????????????,????????????????????? ????:????????????????????????????????????WWW???,???????????????????-??????????????????????,???07-5253809,????????? problem@math.nsysu.edu.tw ??????????????????? ??????????????????????????????????,???????????????????????????????????????????????1?8????????????,??????????????????????????,???????,?????????????????? ????: WWW??(http://www.math.nsysu.edu.tw/~problem) ????????(?4009?)   92???????????????:9/19, 10/3, 10/17, 10/31, 11/14, 11/28, 12/12, 12/26  ????:???????????            ????:??????

    52. The End Thank You

More Related