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Graph Theory 1736Euler

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Graph Theory 1736Euler

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

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

    3. Knigsberg

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

    5. Problem of the Knigsberg 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. Knigsberg 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. Eulers Formula Eulers Formula: v-e+f=2.

    29. History of Eulers 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. ??: (Eulers 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 Eulers 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

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