Reduc ions o he 3-Decomposi ion Conjec u e
Oli e Bach le and I ene Hein ich
Depa men o Ma hema ics
TU Kaise slau e n
and
Depa men o Ma hema ics
TU Da ms ad
XII La in-Ame ican Algo i hms, G aphs and Op imiza ion Symposium,
2023
Mo i a ion
Fou Colou Theo em
E e y plana g aph is 4-colou able.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 1 / 16
Mo i a ion
Fou Colou Theo em
E e y plana g aph is 4-colou able.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 1 / 16
Mo i a ion
Fou Colou Theo em
E e y plana g aph is 4-colou able.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 1 / 16
Mo i a ion
Fou Colou Theo em
E e y plana g aph is 4-colou able.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 1 / 16
Mo i a ion
Fou Colou Theo em
E e y plana g aph is 4-colou able.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 1 / 16
P o ing he Fou Colou Theo em
Find a se o con igu a ions ha is
▶ educible and
▶checked by a compu e
▶una oidable.
▶by hand, 400 pages
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 2 / 16
P o ing he Fou Colou Theo em
Find a se o con igu a ions ha is
▶ educible and
▶checked by a compu e
▶una oidable.
▶by hand, 400 pages
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 2 / 16
P o ing he Fou Colou Theo em
Find a se o con igu a ions ha is
▶ educible and
▶checked by a compu e
▶una oidable.
▶by hand, 400 pages
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 2 / 16
P o ing he Fou Colou Theo em
Find a se o con igu a ions ha is
▶ educible and
▶checked by a compu e
▶una oidable.
▶by hand, 400 pages
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 2 / 16
P o ing he Fou Colou Theo em
Find a se o con igu a ions ha is
▶ educible and
▶checked by a compu e
▶una oidable.
▶by hand, 400 pages
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 2 / 16
Ou line
The 3-Decomposi ion Conjec u e
Reducible Con igu a ions
Una oidable S uc u es
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 3 / 16
The 3-Decomposi ion Conjec u e
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 4 / 16
3-Decomposi ions o G aphs
De ini ion
A g aph is cubic i e e y e ex has deg ee 3.
De ini ion
A3-decomposi ion o a connec ed cubic g aph Gconsis s o
▶aspanning ee T,
▶aunion o cycles C, and
▶ama ching M
such ha E(G)is he disjoin union E(T)∪E(C)∪M.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 5 / 16
3-Decomposi ions o G aphs
De ini ion
A g aph is cubic i e e y e ex has deg ee 3.
De ini ion
A3-decomposi ion o a connec ed cubic g aph Gconsis s o
▶aspanning ee T,
▶aunion o cycles C, and
▶ama ching M
such ha E(G)is he disjoin union E(T)∪E(C)∪M.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 5 / 16
3-Decomposi ions o G aphs
De ini ion
A g aph is cubic i e e y e ex has deg ee 3.
De ini ion
A3-decomposi ion o a connec ed cubic g aph Gconsis s o
▶aspanning ee T,
▶aunion o cycles C, and
▶ama ching M
such ha E(G)is he disjoin union E(T)∪E(C)∪M.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 5 / 16
An Example G aph
▶Gi en: connec ed cubic g aph.
▶Take a spanning ee.
▶The emaining edges o m
cycles and pa hs.
▶Wan pa hs o leng h 1.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 6 / 16
An Example G aph
▶Gi en: connec ed cubic g aph.
▶Take a spanning ee.
▶The emaining edges o m
cycles and pa hs.
▶Wan pa hs o leng h 1.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 6 / 16
An Example G aph
▶Gi en: connec ed cubic g aph.
▶Take a spanning ee.
▶The emaining edges o m
cycles and pa hs.
▶Wan pa hs o leng h 1.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 6 / 16
Reducible Con igu a ions
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 8 / 16
Reducible Con igu a ions
De ini ion
A educible con igu a ion is a g aph ha is no pa o a 3-connec ed
minimum coun e example o he 3-decomposi ion conjec u e.
Example
The iangle is educible.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 9 / 16
Reducible Con igu a ions
De ini ion
A educible con igu a ion is a g aph ha is no pa o a 3-connec ed
minimum coun e example o he 3-decomposi ion conjec u e.
Example
The iangle is educible.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 9 / 16
Reducible Con igu a ions
De ini ion
A educible con igu a ion is a g aph ha is no pa o a 3-connec ed
minimum coun e example o he 3-decomposi ion conjec u e.
Example
The iangle is educible.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 9 / 16
Reducible Con igu a ions
De ini ion
A educible con igu a ion is a g aph ha is no pa o a 3-connec ed
minimum coun e example o he 3-decomposi ion conjec u e.
Example
The iangle is educible.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 9 / 16
Reducible Con igu a ions
De ini ion
A educible con igu a ion is a g aph ha is no pa o a 3-connec ed
minimum coun e example o he 3-decomposi ion conjec u e.
Example
The iangle is educible.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 9 / 16
Reducible Con igu a ions
De ini ion
A educible con igu a ion is a g aph ha is no pa o a 3-connec ed
minimum coun e example o he 3-decomposi ion conjec u e.
Example
The iangle is educible.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 9 / 16
Reducible Con igu a ions
De ini ion
A educible con igu a ion is a g aph ha is no pa o a 3-connec ed
minimum coun e example o he 3-decomposi ion conjec u e.
Example
The iangle is educible.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 9 / 16
Non-Reducibili y o he squa e
Example
The squa e is no educible.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 10 / 16
Non-Reducibili y o he squa e
Example
The squa e is no educible.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 10 / 16
A Bigge Example
Example
The claw-squa e is educible.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 11 / 16
A Bigge Example
Example
The claw-squa e is educible.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 11 / 16
A Lis o Reducible Con igu a ions
Theo em
The g aphs below a e educible con igu a ions.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 12 / 16
A Lis o Reducible Con igu a ions
Theo em
The g aphs below a e educible con igu a ions.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 12 / 16
A Lis o Reducible Con igu a ions
Theo em
The g aphs below a e educible con igu a ions.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 12 / 16
A Lis o Reducible Con igu a ions
Theo em
The g aphs below a e educible con igu a ions.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 12 / 16
A Lis o Reducible Con igu a ions
Theo em
The g aphs below a e educible con igu a ions.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 12 / 16
A Lis o Reducible Con igu a ions
Theo em
The g aphs below a e educible con igu a ions.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 12 / 16
Una oidable S uc u es
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 13 / 16
Una oidable S uc u es
Ques ion: A e hese educible con igu a ions una oidable?
Answe : No (sadly).
Solu ion: Res ic he class o cubic g aphs o make hem una oidable.
⇒Bound he pa h-wid h.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 14 / 16
Bounding he Pa h-Wid h
Theo em
Cubic g aphs o pa h-wid h a mos 4con ain a educible con igu a ion.
Co olla y
E e y 3-connec ed cubic g aph o pa h-wid h a mos 4sa is ies he
3-decomposi ion conjec u e.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 15 / 16
Summa y
▶We ha e seen six educible con igu a ions,
▶no ed ha hey a e una oidable o pa h-wid h 4, and
▶deduced ha he 3-decomposi ion conjec u e holds o 3-connec ed
cubic g aphs o pa h-wid h 4.
Con ac : o.bach le @ma h. p u.de
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 16 / 16
Summa y
▶We ha e seen six educible con igu a ions,
▶no ed ha hey a e una oidable o pa h-wid h 4, and
▶deduced ha he 3-decomposi ion conjec u e holds o 3-connec ed
cubic g aphs o pa h-wid h 4.
Con ac : o.bach le @ma h. p u.de
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 16 / 16
In e es ing Ex ensions
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 1 / 3
Reducibili y o he Domino
C1
C2
H2
H4
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 2 / 3
Wha Is Pa h-Wid h?
Leng hy o de ine.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 3 / 3
Wha Is Pa h-Wid h?
De ini ion (Pa h-Decomposi ions)
▶Le Gbe a g aph,
▶P=p1. . . pna pa h,
▶V={V1,...,Vn}be bags (⊆V(G)).
(P,V)is a pa h-decomposi ion o Gi :
▶E e y ∈V(G)is con ained in some bag.
▶E e y u ∈E(G)is co e ed by some bag.
▶The se {pi: ∈Vi}is connec ed o all ∈V(G).
De ini ion (Pa h-Wid h)
Le Gbe a g aph and (P,V)be a pa h-decomposi ion o G.
▶The wid h o (P,V)is max{|V1|,...,|Vn|} − 1.
▶The pa h-wid h o Gis he minimal wid h o i s pa h-decomposi ions.
Leng hy o de ine.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 3 / 3
Wha Is Pa h-Wid h?
Leng hy o de ine.
O. Bach le and I. Hein ich (RPTU) Reduc ions o he 3-Dec. Conjec u e LAGOS 2023 3 / 3
