on barnette's conjecture - max planck...
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![Page 1: On Barnette's Conjecture - Max Planck Societyresources.mpi-inf.mpg.de/.../ss13/GraphTheory/Barnette.pdfWilliam T. Tutte Tutte's Conjecture William T. Tutte Tutte's Conjecture (1971)](https://reader036.vdocuments.mx/reader036/viewer/2022062414/5edcacdbad6a402d66677151/html5/thumbnails/1.jpg)
On Barnette's Conjecture
Jens M. Schmidt
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Hamiltonian Cycles
Def. A graph is Hamiltonian if it contains a Hamiltonian cycle, i.e., a cycle that contains every vertex exactly once.
William R. Hamilton
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3-Connectivity
Let G=(V,E) be a simple finite graph, n=|V|, m=|E|.
Def. G is 3-connected n > 3 and there is no separation pair in G
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3-Connectivity
Let G=(V,E) be a simple finite graph, n=|V|, m=|E|.
Def. G is 3-connected n > 3 and there is no separation pair in G
Thm (Steinitz 1922). The graphs of convex 3-dimensional polyhedra are exactly the (simple) planar 3-connected graphs.
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Tait's Conjecture
Tait's Conjecture (1884). Every cubic planar 3-connected graph is Hamiltonian.
Peter G. Tait
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Tait's Conjecture
Tait's Conjecture (1884). Every cubic planar 3-connected graph is Hamiltonian.
every vertex is incident to exactly 3 edges
Peter G. Tait
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Tait's Conjecture
Tait's Conjecture (1884). Every cubic planar 3-connected graph is Hamiltonian.
Peter G. Tait
?every vertex is incident to exactly 3 edges
would imply the 4-color theorem
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Tait's Conjecture
Tait's Conjecture (1884). Every cubic planar 3-connected graph is Hamiltonian.
Tutte's “fragment”
William T. Tutte
...
......
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Tait's Conjecture
Tait's Conjecture (1884). Every cubic planar 3-connected graph is Hamiltonian.
Tutte's counterexample (1946, 46 vertices)
William T. Tutte
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Tutte's Conjecture
William T. Tutte
Tutte's Conjecture (1971). Every cubic bipartite 3-connected graph is Hamiltonian.
planar
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Tutte's Conjecture
Tutte's Conjecture (1971). Every cubic bipartite 3-connected graph is Hamiltonian.
William T. Tutte
?
planar
![Page 12: On Barnette's Conjecture - Max Planck Societyresources.mpi-inf.mpg.de/.../ss13/GraphTheory/Barnette.pdfWilliam T. Tutte Tutte's Conjecture William T. Tutte Tutte's Conjecture (1971)](https://reader036.vdocuments.mx/reader036/viewer/2022062414/5edcacdbad6a402d66677151/html5/thumbnails/12.jpg)
Tutte's Conjecture
Tutte's Conjecture (1971). Every cubic bipartite 3-connected graph is Hamiltonian.
Counterexamples of Baraev & Faradzhev (1978) and Horton (1982)
Joseph D. Horton
planar
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Barnette's Conjecture
Barnette's Conjecture (1969). Every cubic bipartite planar 3-connected graph is Hamiltonian.
David W. Barnette
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Barnette's Conjecture
Barnette's Conjecture (1969). Every cubic bipartite planar 3-connected graph is Hamiltonian.
David W. Barnette
?still open...
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Barnette's Conjecture
Barnette's Conjecture (1969). Every cubic bipartite planar 3-connected graph is Hamiltonian.
...but on the edge:
● Thm (Tutte 1956). Every planar 4-connected graph is Hamiltonian.
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Barnette's Conjecture
Barnette's Conjecture (1969). Every cubic bipartite planar 3-connected graph is Hamiltonian.
...but on the edge:
● Thm (Tutte 1956). Every planar 4-connected graph is Hamiltonian.
● Without assuming cubicness, non-Hamiltonian graphs exist.
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Barnette's Conjecture
Barnette's Conjecture (1969). Every cubic bipartite planar 3-connected graph is Hamiltonian.
...but on the edge:
● Thm (Tutte 1956). Every planar 4-connected graph is Hamiltonian.
● Without assuming cubicness, non-Hamiltonian graphs exist.
Kirkman graph
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Barnette's Conjecture
Barnette's Conjecture (1969). Every cubic bipartite planar 3-connected graph is Hamiltonian.
...but on the edge:
● Thm (Tutte 1956). Every planar 4-connected graph is Hamiltonian.
● Without assuming cubicness, non-Hamiltonian graphs exist.
Kirkman graph
“Barnette graph”
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Strengthening the Conjecture
A Barnette graph is x-y-Hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y.
Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-Hamiltonian.
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Strengthening the Conjecture
A Barnette graph is x-y-Hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y.
Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-Hamiltonian.
Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-Hamiltonian where xyz is a path of length 3.
z
y
x
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Strengthening the Conjecture
A Barnette graph is x-y-Hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y.
Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-Hamiltonian.
Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-Hamiltonian where xyz is a path of length 3.
Proof.● <=: easy
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Strengthening the Conjecture
A Barnette graph is x-y-Hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y.
Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-Hamiltonian.
Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-Hamiltonian where xyz is a path of length 3.
Proof.● <=: easy● =>:
z
x
y
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Strengthening the Conjecture
A Barnette graph is x-y-Hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y.
Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-Hamiltonian.
Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-Hamiltonian where xyz is a path of length 3.
Proof.● <=: easy● =>:
x‘ y‘
![Page 24: On Barnette's Conjecture - Max Planck Societyresources.mpi-inf.mpg.de/.../ss13/GraphTheory/Barnette.pdfWilliam T. Tutte Tutte's Conjecture William T. Tutte Tutte's Conjecture (1971)](https://reader036.vdocuments.mx/reader036/viewer/2022062414/5edcacdbad6a402d66677151/html5/thumbnails/24.jpg)
Strengthening the Conjecture
A Barnette graph is x-y-Hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y.
Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-Hamiltonian.
Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-Hamiltonian where xyz is a path of length 3.
Proof.● <=: easy● =>:
x‘ y‘
![Page 25: On Barnette's Conjecture - Max Planck Societyresources.mpi-inf.mpg.de/.../ss13/GraphTheory/Barnette.pdfWilliam T. Tutte Tutte's Conjecture William T. Tutte Tutte's Conjecture (1971)](https://reader036.vdocuments.mx/reader036/viewer/2022062414/5edcacdbad6a402d66677151/html5/thumbnails/25.jpg)
Strengthening the Conjecture
A Barnette graph is x-y-Hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y.
Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-Hamiltonian.
Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-Hamiltonian where xyz is a path of length 3.
Proof.● <=: easy● =>:
x‘ y‘
![Page 26: On Barnette's Conjecture - Max Planck Societyresources.mpi-inf.mpg.de/.../ss13/GraphTheory/Barnette.pdfWilliam T. Tutte Tutte's Conjecture William T. Tutte Tutte's Conjecture (1971)](https://reader036.vdocuments.mx/reader036/viewer/2022062414/5edcacdbad6a402d66677151/html5/thumbnails/26.jpg)
Strengthening the Conjecture
A Barnette graph is x-y-Hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y.
Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-Hamiltonian.
Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-Hamiltonian where xyz is a path of length 3.
Proof.● <=: easy● =>:
![Page 27: On Barnette's Conjecture - Max Planck Societyresources.mpi-inf.mpg.de/.../ss13/GraphTheory/Barnette.pdfWilliam T. Tutte Tutte's Conjecture William T. Tutte Tutte's Conjecture (1971)](https://reader036.vdocuments.mx/reader036/viewer/2022062414/5edcacdbad6a402d66677151/html5/thumbnails/27.jpg)
Strengthening the Conjecture
A Barnette graph is x-y-Hamiltonian if for every two edges x and y in a common face f there is a Hamiltonian cycle that contains x but not y.
Thm (Kelmans 1986). Barnette's conjecture is true if and only if every Barnette graph is x-y-Hamiltonian.
Thm (Hertel 2005). Barnette's conjecture is true if and only if every Barnette graph is x-y-z-Hamiltonian where xyz is a path of length 3.
Proof.● <=: easy● =>:
y
![Page 28: On Barnette's Conjecture - Max Planck Societyresources.mpi-inf.mpg.de/.../ss13/GraphTheory/Barnette.pdfWilliam T. Tutte Tutte's Conjecture William T. Tutte Tutte's Conjecture (1971)](https://reader036.vdocuments.mx/reader036/viewer/2022062414/5edcacdbad6a402d66677151/html5/thumbnails/28.jpg)
Dualization & Computer-aided Search
Searching for counterexamples:
Thm (Brinkmann, McKay 2002). Every Barnette graph with at most 84 vertices is Hamiltonian.
Thm (Brinkmann, McKay 2002). Every Barnette graph with at most 60 vertices is x-y-Hamiltonian. Even true if x and y are not necessarily in the same face.
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Weakening the Conjecture
Def. A graph is cyclically 4-edge-connected if every cycle-separating edge cut contains at least 4 edges.
not cyclically 4-edge-connected cyclically 4-edge-connected
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Weakening the Conjecture
Thm (S.). Barnette's conjecture is true if and only if every cyclically 4-edge-connected Barnette graph is x-y-Hamiltonian.
Proof.● =>: easy.
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Weakening the Conjecture
Thm (S.). Barnette's conjecture is true if and only if every cyclically 4-edge-connected Barnette graph is x-y-Hamiltonian.
Proof.● =>: easy.● <=: Take any non-cyclically 4-edge-connected Barnette graph.
BA
![Page 32: On Barnette's Conjecture - Max Planck Societyresources.mpi-inf.mpg.de/.../ss13/GraphTheory/Barnette.pdfWilliam T. Tutte Tutte's Conjecture William T. Tutte Tutte's Conjecture (1971)](https://reader036.vdocuments.mx/reader036/viewer/2022062414/5edcacdbad6a402d66677151/html5/thumbnails/32.jpg)
Weakening the Conjecture
Thm (S.). Barnette's conjecture is true if and only if every cyclically 4-edge-connected Barnette graph is x-y-Hamiltonian.
Proof.● =>: easy.● <=: Take any non-cyclically 4-edge-connected Barnette graph.
BA
- cubic
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Weakening the Conjecture
Thm (S.). Barnette's conjecture is true if and only if every cyclically 4-edge-connected Barnette graph is x-y-Hamiltonian.
Proof.● =>: easy.● <=: Take any non-cyclically 4-edge-connected Barnette graph.
BA
- cubic- simple
![Page 34: On Barnette's Conjecture - Max Planck Societyresources.mpi-inf.mpg.de/.../ss13/GraphTheory/Barnette.pdfWilliam T. Tutte Tutte's Conjecture William T. Tutte Tutte's Conjecture (1971)](https://reader036.vdocuments.mx/reader036/viewer/2022062414/5edcacdbad6a402d66677151/html5/thumbnails/34.jpg)
Weakening the Conjecture
Thm (S.). Barnette's conjecture is true if and only if every cyclically 4-edge-connected Barnette graph is x-y-Hamiltonian.
Proof.● =>: easy.● <=: Take any non-cyclically 4-edge-connected Barnette graph.
BA
- cubic- simple- 3-connected
![Page 35: On Barnette's Conjecture - Max Planck Societyresources.mpi-inf.mpg.de/.../ss13/GraphTheory/Barnette.pdfWilliam T. Tutte Tutte's Conjecture William T. Tutte Tutte's Conjecture (1971)](https://reader036.vdocuments.mx/reader036/viewer/2022062414/5edcacdbad6a402d66677151/html5/thumbnails/35.jpg)
Weakening the Conjecture
Thm (S.). Barnette's conjecture is true if and only if every cyclically 4-edge-connected Barnette graph is x-y-Hamiltonian.
Proof.● =>: easy.● <=: Take any non-cyclically 4-edge-connected Barnette graph.
- cubic- simple- 3-connected- bipartite
BA
![Page 36: On Barnette's Conjecture - Max Planck Societyresources.mpi-inf.mpg.de/.../ss13/GraphTheory/Barnette.pdfWilliam T. Tutte Tutte's Conjecture William T. Tutte Tutte's Conjecture (1971)](https://reader036.vdocuments.mx/reader036/viewer/2022062414/5edcacdbad6a402d66677151/html5/thumbnails/36.jpg)
Weakening the Conjecture
Thm (S.). Barnette's conjecture is true if and only if every cyclically 4-edge-connected Barnette graph is x-y-Hamiltonian.
Proof.● =>: easy.● <=: Take any non-cyclically 4-edge-connected Barnette graph.
- cubic- simple- 3-connected- bipartite
BA
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Special Faces
Thm (Goodey 1975). Every Barnette graph that contains only faces of degree 4 and 6 (and at most one special face of degree 8) is Hamiltonian.
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Special Faces
Thm (Goodey 1975). Every Barnette graph that contains only faces of degree 4 and 6 (and at most one special face of degree 8) is Hamiltonian.
Thm (Feder, Subi 2009). Every Barnette graph that contains only faces of degree 4 and 6 for two of the three face-color classes is Hamiltonian.
![Page 39: On Barnette's Conjecture - Max Planck Societyresources.mpi-inf.mpg.de/.../ss13/GraphTheory/Barnette.pdfWilliam T. Tutte Tutte's Conjecture William T. Tutte Tutte's Conjecture (1971)](https://reader036.vdocuments.mx/reader036/viewer/2022062414/5edcacdbad6a402d66677151/html5/thumbnails/39.jpg)
Special Faces
Thm (Goodey 1975). Every Barnette graph that contains only faces of degree 4 and 6 (and at most one special face of degree 8) is Hamiltonian.
Thm (Feder, Subi 2009). Every Barnette graph that contains only faces of degree 4 and 6 for two of the three face-color classes is Hamiltonian.
Computing a Hamiltonian cycle for every such graph is in P.
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End of the first talk. Don't run away.