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Every 4-connected graph with crossing number 2 is hamiltonian
Joint work with Carol Zamfirescu (Ghent University, Belgium)
TGT 30 Date: 24 (Wed.) 26 (Fri.), October 2018 Place: Hatoba Hall, - - PowerPoint PPT Presentation
TGT 30 Date: 24 (Wed.) 26 (Fri.), October 2018 Place: Hatoba Hall, - - PowerPoint PPT Presentation
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
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Hamiltonicity of plane graphs
Hamilton cycle in a graph A cycle visiting all vertices
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Hamiltonicity of plane graphs
Tait (1884) : Hamiltonian cycle in cubic map 4-coloring in plane graph Hamilton cycle in a graph A cycle visiting all vertices
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Hamiltonicity of plane graphs
False Tait (1884) : Hamiltonian cycle in cubic map 4-coloring in plane graph Hamilton cycle in a graph A cycle visiting all vertices
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Hamiltonicity of plane graphs
False True (4-color thm.) Tait (1884) : Hamiltonian cycle in cubic map 4-coloring in plane graph Hamilton cycle in a graph A cycle visiting all vertices
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Hamiltonicity of plane graphs
4-connected plane graph has a Hamilton cycle
- Thm. (Tutte, `56)
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Hamiltonicity of plane graphs
Many works for graphs on surfaces
4-connected plane graph has a Hamilton cycle
- Thm. (Tutte, `56)
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Projective plane
Hamiltonicity of plane graphs
Many works for graphs on surfaces
4-connected plane graph has a Hamilton cycle
- Thm. (Tutte, `56)
✓
Thomas & Yu `94
✓
K.K. & Oz. `14
✓
Thomassen `83
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Projective plane Torus
Hamiltonicity of plane graphs
Many works for graphs on surfaces
4-connected plane graph has a Hamilton cycle
- Thm. (Tutte, `56)
✓
Thomas & Yu `94
✓
K.K. & Oz. `14
✓
Thomas, Yu & Zang `05
✓
K.K. & Oz. `16
✓
Thomassen `83
✓
Thomas & Yu `97
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Projective plane Torus K-bottle
Hamiltonicity of plane graphs
Many works for graphs on surfaces
4-connected plane graph has a Hamilton cycle
- Thm. (Tutte, `56)
✓
Thomas & Yu `94
✓
K.K. & Oz. `14
✓
Thomas, Yu & Zang `05
✓
K.K. & Oz. `16
✓
Brunet, Nakamoto & Negami `99
✓
Thomassen `83
✓
Thomas & Yu `97
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Crossing number and Hamiltonicity
We study this from another aspect, crossing number Many works for graphs on surfaces
4-connected plane graph has a Hamilton cycle
- Thm. (Tutte, `56)
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Crossing number and Hamiltonicity
We study this from another aspect, crossing number Many works for graphs on surfaces
4-connected plane graph has a Hamilton cycle
- Thm. (Tutte, `56)
crossing
G : graph
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Crossing number and Hamiltonicity
We study this from another aspect, crossing number Many works for graphs on surfaces
4-connected plane graph has a Hamilton cycle
- Thm. (Tutte, `56)
crossing
G : graph Consider drawing of G with min. # of crossings
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Crossing number and Hamiltonicity
We study this from another aspect, crossing number Many works for graphs on surfaces
4-connected plane graph has a Hamilton cycle
- Thm. (Tutte, `56)
crossing
G : graph Consider drawing of G cr(G) : # of its crossings with min. # of crossings
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The case of small crossing number
◼ cr(G) = 1
G : projective planar
Projective plane
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The case of small crossing number
◼ cr(G) = 1
G : projective planar
Projective plane
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◼ cr(G) = 1
G : projective planar 4-conn. graph G with cr(G) = 1 has a Hamilton cycle
- Cor. of Thomas & Yu, `94
The case of small crossing number
Projective plane
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◼ cr(G) = 2
G : embeddable on K-bottle
◼ cr(G) = 1
G : projective planar 4-conn. graph G with cr(G) = 1 has a Hamilton cycle
- Cor. of Thomas & Yu, `94
Projective plane K-bottle
The case of small crossing number
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◼ 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
◼ cr(G) = 1
G : projective planar 4-conn. graph G with cr(G) = 1 has a Hamilton cycle
- Cor. of Thomas & Yu, `94
The case of small crossing number
Projective plane K-bottle
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4-conn. graph G with cr(G) = 2 has a Hamilton cycle
- Thm. ( Oz. & Zamfirescu `17+)
The case of small crossing number
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4-conn. graph G with cr(G) = 2 has a Hamilton cycle
- Thm. ( Oz. & Zamfirescu `17+)
4-conn. graph G with cr(G) = 6 and no Hamilton cycle
Prop.
The case of small crossing number
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4-conn. graph G with cr(G) = 2 has a Hamilton cycle
- Thm. ( Oz. & Zamfirescu `17+)
4-conn. graph G with cr(G) = 6 and no Hamilton cycle
Prop.
What about 4-conn. graphs G with cr(G) = 3, 4, 5?
The case of small crossing number
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Hamiltonicity and 1-tough
◼ G has a Hamilton cycle
G : 1-tough
S
What about 4-conn. graphs G with cr(G) = 3, 4, 5? S : cutset, (# of comp.s of G－S)
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Hamiltonicity and 1-tough
◼ G has a Hamilton cycle
G : 1-tough
S
What about 4-conn. graphs G with cr(G) = 3, 4, 5? S : cutset, (# of comp.s of G－S)
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Hamiltonicity and 1-tough
◼ G has a Hamilton cycle
G : 1-tough S : cutset, (# of comp.s of G－S)
S
4-conn. graph G with cr(G) is 1-tough
Prop.
What about 4-conn. graphs G with cr(G) = 3, 4, 5?
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Crossing number 2
4-conn. graph G with cr(G) = 2 has a Hamilton cycle
- Thm. ( Oz. & Zamfirescu `17+)
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Crossing number 2
Proof: Add a new vertex on the 2 crossing points
crossing
graph G 4-conn. graph G with cr(G) = 2 has a Hamilton cycle
- Thm. ( Oz. & Zamfirescu `17+)
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Crossing number 2
Proof: Add a new vertex on the 2 crossing points
crossing
graph G
New vertex
Plane graph 4-conn. graph G with cr(G) = 2 has a Hamilton cycle
- Thm. ( Oz. & Zamfirescu `17+)
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Crossing number 2
If is 4-conn. Hamilton cycle
crossing
graph G
New vertex
Plane graph
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Crossing number 2
If is 4-conn. Hamilton cycle For G : 4-conn. planar and , has a Hamilton cycle. (Thomas & Yu, `94)
crossing
graph G
New vertex
Plane graph
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Crossing number 2
So, : NOT 4-conn. If is 4-conn. Hamilton cycle For G : 4-conn. planar and , has a Hamilton cycle. (Thomas & Yu, `94)
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Crossing number 2
So, : NOT 4-conn. Since G : 4-conn.,
crossing
4-cut as in the right figure If is 4-conn. Hamilton cycle For G : 4-conn. planar and , has a Hamilton cycle. (Thomas & Yu, `94)
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Crossing number 2
crossing Plane graph crossing
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Crossing number 2
Plane graph crossing
: 4-connected crossing # = 1
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Crossing number 2
Hamilton cycle in (without edge-crossing) : 4-connected crossing # = 1
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Crossing number 2
Hamilton cycle in (without edge-crossing) Modify it suitably : 4-connected crossing # = 1
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Crossing number 2
Hamilton cycle in (without edge-crossing) Modify it suitably : 4-connected crossing # = 1
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Crossing number 2
Modify it suitably : 4-connected crossing # = 1 Hamilton cycle in (without edge-crossing)
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Crossing number 2
Modify it suitably : 4-connected crossing # = 1 Hamilton cycle in (without edge-crossing)
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Crossing number 2
Modify it suitably : 4-connected crossing # = 1 Hamilton cycle in (without edge-crossing)
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Crossing number 2
Modify it suitably : 4-connected crossing # = 1 Hamilton cycle in (without edge-crossing)
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Crossing number 2
Modify it suitably : 4-connected crossing # = 1 Hamilton cycle in (without edge-crossing)
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Crossing number 2
Modify it suitably
??
: 4-connected crossing # = 1 Hamilton cycle in (without edge-crossing)
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Crossing number 2
Modify the right part! : 4-connected crossing # = 1
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Crossing number 2
Modify the right part! Add an edge e as above, e : 4-connected crossing # = 1
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Crossing number 2
Add an edge e as above, and find a H-cycle thr. e : 4-connected crossing # = 1 e Modify the right part!
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Crossing number 2
e : 4-connected crossing # = 1 Add an edge e as above, and find a H-cycle thr. e Modify the right part!
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Crossing number 2
e : 4-connected crossing # = 1 Add an edge e as above, and find a H-cycle thr. e Modify the right part!
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Summary
4-conn. graph G with cr(G) = 2 has a Hamilton cycle
- Thm. ( Oz. & Zamfirescu `17+)
4-conn. graph G with cr(G) = 6 and no Hamilton cycle
Prop.
What about 4-conn. graphs G with cr(G) = 3, 4, 5?
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Thank you for your attention
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