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TGT 30 Date: 24 (Wed.) 26 (Fri.), October 2018 Place: Hatoba Hall, - PowerPoint PPT Presentation

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TGT 30 Date: 24 (Wed.) 26 (Fri.), October 2018 Place: Hatoba Hall, Yokohama Here 24th May, 2018 JCCA2018 1 Every 4-connected graph with crossing number 2 is hamiltonian Kenta Ozeki (Yokohama National Univeristy) Joint work with Carol

  1. TGT 30 Date: 24 (Wed.) – 26 (Fri.), October 2018 Place: Hatoba Hall, Yokohama Here 24th May, 2018 JCCA2018 1

  2. Every 4-connected graph with crossing number 2 is hamiltonian Kenta Ozeki (Yokohama National Univeristy) Joint work with Carol Zamfirescu (Ghent University, Belgium)

  3. Hamiltonicity of plane graphs Hamilton cycle in a graph A cycle visiting all vertices 24th May, 2018 JCCA2018 3

  4. Hamiltonicity of plane graphs Hamilton cycle in a graph A cycle visiting all vertices Tait (1884) : Hamiltonian cycle in cubic map 4-coloring in plane graph 24th May, 2018 JCCA2018 4

  5. Hamiltonicity of plane graphs Hamilton cycle in a graph A cycle visiting all vertices Tait (1884) : Hamiltonian cycle in cubic map False 4-coloring in plane graph 24th May, 2018 JCCA2018 5

  6. Hamiltonicity of plane graphs Hamilton cycle in a graph A cycle visiting all vertices Tait (1884) : Hamiltonian cycle in cubic map False 4-coloring in plane graph True (4-color thm . ) 24th May, 2018 JCCA2018 6

  7. Hamiltonicity of plane graphs Thm. (Tutte, `56) 4-connected plane graph has a Hamilton cycle 24th May, 2018 JCCA2018 7

  8. Hamiltonicity of plane graphs Thm. (Tutte, `56) 4-connected plane graph has a Hamilton cycle Many works for graphs on surfaces 24th May, 2018 JCCA2018 8

  9. Hamiltonicity of plane graphs Thm. (Tutte, `56) 4-connected plane graph has a Hamilton cycle Many works for graphs on surfaces Projective plane Thomassen `83 ✓ Thomas & Yu `94 ✓ K.K. & Oz. `14 ✓ 24th May, 2018 JCCA2018 9

  10. Hamiltonicity of plane graphs Thm. (Tutte, `56) 4-connected plane graph has a Hamilton cycle Many works for graphs on surfaces Torus Projective plane Thomassen `83 ✓ Thomas & Yu `97 ✓ Thomas & Yu `94 Thomas, Yu & Zang `05 ✓ ✓ K.K. & Oz. `14 K.K. & Oz. `16 ✓ ✓ 24th May, 2018 JCCA2018 10

  11. Hamiltonicity of plane graphs Thm. (Tutte, `56) 4-connected plane graph has a Hamilton cycle Many works for graphs on surfaces Torus Projective plane K-bottle Thomassen `83 Brunet, Nakamoto ✓ ✓ Thomas & Yu `97 ✓ Thomas & Yu `94 & Negami `99 Thomas, Yu & Zang `05 ✓ ✓ K.K. & Oz. `14 K.K. & Oz. `16 ✓ ✓ 24th May, 2018 JCCA2018 11

  12. Crossing number and Hamiltonicity Thm. (Tutte, `56) 4-connected plane graph has a Hamilton cycle Many works for graphs on surfaces We study this from another aspect, crossing number 24th May, 2018 JCCA2018 12

  13. Crossing number and Hamiltonicity Thm. (Tutte, `56) 4-connected plane graph has a Hamilton cycle Many works for graphs on surfaces We study this from another aspect, crossing number G : graph crossing 24th May, 2018 JCCA2018 13

  14. Crossing number and Hamiltonicity Thm. (Tutte, `56) 4-connected plane graph has a Hamilton cycle Many works for graphs on surfaces We study this from another aspect, crossing number G : graph crossing Consider drawing of G with min. # of crossings 24th May, 2018 JCCA2018 14

  15. Crossing number and Hamiltonicity Thm. (Tutte, `56) 4-connected plane graph has a Hamilton cycle Many works for graphs on surfaces We study this from another aspect, crossing number G : graph crossing Consider drawing of G with min. # of crossings cr( G ) : # of its crossings 24th May, 2018 JCCA2018 15

  16. The case of small crossing number ◼ cr( G ) = 1 G : projective planar Projective plane 24th May, 2018 JCCA2018 16

  17. The case of small crossing number ◼ cr( G ) = 1 G : projective planar Projective plane 24th May, 2018 JCCA2018 17

  18. The case of small crossing number ◼ cr( G ) = 1 G : projective planar Cor. of Thomas & Yu, `94 4-conn. graph G with cr(G) = 1 has a Hamilton cycle Projective plane 24th May, 2018 JCCA2018 18

  19. The case of small crossing number ◼ cr( G ) = 1 G : projective planar Cor. of Thomas & Yu, `94 4-conn. graph G with cr(G) = 1 has a Hamilton cycle K-bottle Projective plane ◼ cr( G ) = 2 G : embeddable on K-bottle 24th May, 2018 JCCA2018 19

  20. The case of small crossing number ◼ cr( G ) = 1 G : projective planar Cor. of Thomas & Yu, `94 4-conn. graph G with cr(G) = 1 has a Hamilton cycle K-bottle Projective plane ◼ cr( G ) = 2 G : embeddable on K-bottle Does 4-conn. graph on K-bottle have a Hamilton cycle? c.f. Conj. for torus by Grunbaum `70, Nash-Williams `73 24th May, 2018 JCCA2018 20

  21. The case of small crossing number Thm. ( Oz. & Zamfirescu `17+) 4-conn. graph G with cr( G ) = 2 has a Hamilton cycle 24th May, 2018 JCCA2018 21

  22. The case of small crossing number Thm. ( Oz. & Zamfirescu `17+) 4-conn. graph G with cr( G ) = 2 has a Hamilton cycle Prop. 4-conn. graph G with cr( G ) = 6 and no Hamilton cycle 24th May, 2018 JCCA2018 22

  23. The case of small crossing number Thm. ( Oz. & Zamfirescu `17+) 4-conn. graph G with cr( G ) = 2 has a Hamilton cycle Prop. 4-conn. graph G with cr( G ) = 6 and no Hamilton cycle What about 4-conn. graphs G with cr( G ) = 3, 4, 5? 24th May, 2018 JCCA2018 23

  24. Hamiltonicity and 1-tough ◼ G has a Hamilton cycle G : 1-tough S S : cutset, (# of comp.s of G － S ) What about 4-conn. graphs G with cr( G ) = 3, 4, 5? 24th May, 2018 JCCA2018 24

  25. Hamiltonicity and 1-tough ◼ G has a Hamilton cycle G : 1-tough S S : cutset, (# of comp.s of G － S ) What about 4-conn. graphs G with cr( G ) = 3, 4, 5? 24th May, 2018 JCCA2018 25

  26. Hamiltonicity and 1-tough ◼ G has a Hamilton cycle G : 1-tough S S : cutset, (# of comp.s of G － S ) Prop. 4-conn. graph G with cr( G ) is 1-tough What about 4-conn. graphs G with cr( G ) = 3, 4, 5? 24th May, 2018 JCCA2018 26

  27. Crossing number 2 Thm. ( Oz. & Zamfirescu `17+) 4-conn. graph G with cr( G ) = 2 has a Hamilton cycle 24th May, 2018 JCCA2018 27

  28. Crossing number 2 Thm. ( Oz. & Zamfirescu `17+) 4-conn. graph G with cr( G ) = 2 has a Hamilton cycle Proof: Add a new vertex on the 2 crossing points crossing graph G 24th May, 2018 JCCA2018 28

  29. Crossing number 2 Thm. ( Oz. & Zamfirescu `17+) 4-conn. graph G with cr( G ) = 2 has a Hamilton cycle Proof: Add a new vertex on the 2 crossing points New vertex crossing graph G Plane graph 24th May, 2018 JCCA2018 29

  30. Crossing number 2 If is 4-conn. Hamilton cycle New vertex crossing graph G Plane graph 24th May, 2018 JCCA2018 30

  31. Crossing number 2 If is 4-conn. Hamilton cycle For G : 4-conn. planar and , has a Hamilton cycle. (Thomas & Yu, `94) New vertex crossing graph G Plane graph 24th May, 2018 JCCA2018 31

  32. Crossing number 2 If is 4-conn. Hamilton cycle For G : 4-conn. planar and , has a Hamilton cycle. (Thomas & Yu, `94) So, : NOT 4-conn. 24th May, 2018 JCCA2018 32

  33. Crossing number 2 If is 4-conn. Hamilton cycle For G : 4-conn. planar and , has a Hamilton cycle. (Thomas & Yu, `94) So, : NOT 4-conn. Since G : 4-conn., 4-cut as in the right figure crossing 24th May, 2018 JCCA2018 33

  34. Crossing number 2 crossing Plane graph crossing 24th May, 2018 JCCA2018 34

  35. Crossing number 2 crossing Plane graph : 4-connected crossing # = 1 24th May, 2018 JCCA2018 35

  36. Crossing number 2 Hamilton cycle in (without edge-crossing) : 4-connected crossing # = 1 24th May, 2018 JCCA2018 36

  37. Crossing number 2 Hamilton cycle in (without edge-crossing) Modify it suitably : 4-connected crossing # = 1 24th May, 2018 JCCA2018 37

  38. Crossing number 2 Hamilton cycle in (without edge-crossing) Modify it suitably : 4-connected crossing # = 1 24th May, 2018 JCCA2018 38

  39. Crossing number 2 Hamilton cycle in (without edge-crossing) Modify it suitably : 4-connected crossing # = 1 24th May, 2018 JCCA2018 39

  40. Crossing number 2 Hamilton cycle in (without edge-crossing) Modify it suitably : 4-connected crossing # = 1 24th May, 2018 JCCA2018 40

  41. Crossing number 2 Hamilton cycle in (without edge-crossing) Modify it suitably : 4-connected crossing # = 1 24th May, 2018 JCCA2018 41

  42. Crossing number 2 Hamilton cycle in (without edge-crossing) Modify it suitably : 4-connected crossing # = 1 24th May, 2018 JCCA2018 42

  43. Crossing number 2 Hamilton cycle in (without edge-crossing) Modify it suitably : 4-connected crossing # = 1 24th May, 2018 JCCA2018 43

  44. Crossing number 2 Hamilton cycle in (without edge-crossing) Modify it suitably : 4-connected crossing # = 1 ?? 24th May, 2018 JCCA2018 44

  45. Crossing number 2 Modify the right part! : 4-connected crossing # = 1 24th May, 2018 JCCA2018 45

  46. Crossing number 2 Modify the right part! e : 4-connected Add an edge e as above, crossing # = 1 24th May, 2018 JCCA2018 46

  47. Crossing number 2 Modify the right part! e : 4-connected Add an edge e as above, crossing # = 1 and find a H-cycle thr. e 24th May, 2018 JCCA2018 47

  48. Crossing number 2 Modify the right part! e : 4-connected Add an edge e as above, crossing # = 1 and find a H-cycle thr. e 24th May, 2018 JCCA2018 48

  49. Crossing number 2 Modify the right part! e : 4-connected Add an edge e as above, crossing # = 1 and find a H-cycle thr. e 24th May, 2018 JCCA2018 49

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