Date: 24 (Wed.) 26 (Fri.), October 2018 Place: Hatoba Hall ...
Transcript of Date: 24 (Wed.) 26 (Fri.), October 2018 Place: Hatoba Hall ...
24th May, 2018 JCCA2018 1
TGT 30Date: 24 (Wed.) – 26 (Fri.), October 2018
Place: Hatoba Hall, Yokohama Here
Every 4-connected graph with
crossing number 2 is hamiltonian
Joint work with
Carol Zamfirescu (Ghent University, Belgium)
Kenta Ozeki (Yokohama National Univeristy)
24th May, 2018 JCCA2018 3
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
24th May, 2018 JCCA2018 5
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 TorusK-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
24th May, 2018 JCCA2018 15
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
Planegraph
crossing
: 4-connected
crossing # = 1
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Crossing number 2
Hamilton cycle in
(without edge-crossing)
: 4-connected
crossing # = 1
24th May, 2018 JCCA2018 37
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?