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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
24th May, 2018 JCCA2018 4
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
24th May, 2018 JCCA2018 6
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
24th May, 2018 JCCA2018 7
Hamiltonicity of plane graphs
4-connected plane graph has a Hamilton cycle
Thm. (Tutte, `56)
24th May, 2018 JCCA2018 8
Hamiltonicity of plane graphs
Many works for graphs on surfaces
4-connected plane graph has a Hamilton cycle
Thm. (Tutte, `56)
24th May, 2018 JCCA2018 9
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
24th May, 2018 JCCA2018 10
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
24th May, 2018 JCCA2018 11
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
24th May, 2018 JCCA2018 12
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)
24th May, 2018 JCCA2018 13
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
24th May, 2018 JCCA2018 14
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
24th May, 2018 JCCA2018 16
The case of small crossing number
◼ cr(G) = 1 G : projective planar
Projective plane
24th May, 2018 JCCA2018 17
The case of small crossing number
◼ cr(G) = 1 G : projective planar
Projective plane
24th May, 2018 JCCA2018 18
◼ 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
24th May, 2018 JCCA2018 19
◼ 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
24th May, 2018 JCCA2018 20
◼ 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
24th May, 2018 JCCA2018 21
4-conn. graph G with cr(G) = 2 has a Hamilton cycle
Thm. ( Oz. & Zamfirescu `17+)
The case of small crossing number
24th May, 2018 JCCA2018 22
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
24th May, 2018 JCCA2018 23
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
24th May, 2018 JCCA2018 24
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)
24th May, 2018 JCCA2018 25
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)
24th May, 2018 JCCA2018 26
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?
24th May, 2018 JCCA2018 27
Crossing number 2
4-conn. graph G with cr(G) = 2 has a Hamilton cycle
Thm. ( Oz. & Zamfirescu `17+)
24th May, 2018 JCCA2018 28
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+)
24th May, 2018 JCCA2018 29
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+)
24th May, 2018 JCCA2018 30
Crossing number 2
If is 4-conn. Hamilton cycle
crossing
graph G
New vertex
Plane graph
24th May, 2018 JCCA2018 31
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
24th May, 2018 JCCA2018 32
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)
24th May, 2018 JCCA2018 33
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)
24th May, 2018 JCCA2018 34
Crossing number 2
crossing
Planegraph
crossing
24th May, 2018 JCCA2018 35
Crossing number 2
Planegraph
crossing
: 4-connected
crossing # = 1
24th May, 2018 JCCA2018 36
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
24th May, 2018 JCCA2018 38
Crossing number 2
Hamilton cycle in
(without edge-crossing)
Modify it suitably
: 4-connected
crossing # = 1
24th May, 2018 JCCA2018 39
Crossing number 2
Modify it suitably
: 4-connected
crossing # = 1
Hamilton cycle in
(without edge-crossing)
24th May, 2018 JCCA2018 40
Crossing number 2
Modify it suitably
: 4-connected
crossing # = 1
Hamilton cycle in
(without edge-crossing)
24th May, 2018 JCCA2018 41
Crossing number 2
Modify it suitably
: 4-connected
crossing # = 1
Hamilton cycle in
(without edge-crossing)
24th May, 2018 JCCA2018 42
Crossing number 2
Modify it suitably
: 4-connected
crossing # = 1
Hamilton cycle in
(without edge-crossing)
24th May, 2018 JCCA2018 43
Crossing number 2
Modify it suitably
: 4-connected
crossing # = 1
Hamilton cycle in
(without edge-crossing)
24th May, 2018 JCCA2018 44
Crossing number 2
Modify it suitably
??
: 4-connected
crossing # = 1
Hamilton cycle in
(without edge-crossing)
24th May, 2018 JCCA2018 45
Crossing number 2
Modify the right part!
: 4-connected
crossing # = 1
24th May, 2018 JCCA2018 46
Crossing number 2
Modify the right part!
Add an edge e as above,
e
: 4-connected
crossing # = 1
24th May, 2018 JCCA2018 47
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!
24th May, 2018 JCCA2018 48
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!
24th May, 2018 JCCA2018 49
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!
24th May, 2018 JCCA2018 50
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?
Thank you for your attention
24th May, 2018 JCCA2018 52
TGT 30Date: 24 (Wed.) – 26 (Fri.), October 2018
Place: Hatoba Hall, Yokohama Here
