Decomposing Star-Like Cubic Graphs

Decomposing S a -like Cubic G aphs
Oli e Bach le , S en O. K umke
TU Kaise slau e n
Young Ma hema icians Symposium o he G ea e Region, 2020
Ou line
Basics
3-Decomposi ions
Exis ence o Hamil onian G aphs
Cons uc ing a Decomposi ion
F om he Cen e
To he Tips
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 1 / 15
3-Decomposi ions o G aphs
De ini ion (Cubic G aphs)
A g aph is cubic i e e y e ex has deg ee 3.
De ini ion (3-Decomposi ion)
A3-decomposi ion o a connec ed cubic g aph Gconsis s o
Ia spanning ee T,
Ia se o cycles C, and
Ia ma ching M
such ha E(G)is he disjoin union E(T)∪E(C)∪M.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 2 / 15
3-Decomposi ions o G aphs
De ini ion (Cubic G aphs)
A g aph is cubic i e e y e ex has deg ee 3.
De ini ion (3-Decomposi ion)
A3-decomposi ion o a connec ed cubic g aph Gconsis s o
Ia spanning ee T,
Ia se o cycles C, and
Ia ma ching M
such ha E(G)is he disjoin union E(T)∪E(C)∪M.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 2 / 15
3-Decomposi ions o G aphs
De ini ion (Cubic G aphs)
A g aph is cubic i e e y e ex has deg ee 3.
De ini ion (3-Decomposi ion)
A3-decomposi ion o a connec ed cubic g aph Gconsis s o
Ia spanning ee T,
Ia se o cycles C, and
Ia ma ching M
such ha E(G)is he disjoin union E(T)∪E(C)∪M.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 2 / 15

An Example G aph
IGi en: connec ed cubic g aph.
ITake a spanning ee.
IThe emaining edges o m
cycles and pa hs.
IWan pa hs o leng h 1.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 3 / 15
An Example G aph
IGi en: connec ed cubic g aph.
ITake a spanning ee.
IThe emaining edges o m
cycles and pa hs.
IWan pa hs o leng h 1.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 3 / 15
An Example G aph
IGi en: connec ed cubic g aph.
ITake a spanning ee.
IThe emaining edges o m
cycles and pa hs.
IWan pa hs o leng h 1.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 3 / 15
An Example G aph
IGi en: connec ed cubic g aph.
ITake a spanning ee.
IThe emaining edges o m
cycles and pa hs.
IWan pa hs o leng h 1.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 3 / 15
Ou Resul
De ini ion (S a -like G aphs)
Le Gbe a connec ed cubic g aph wi h a pe ec ma ching M.
IG−Mconsis s o cycles.
ICon ac ing hese yields he con ac ion g aph GM.
Gis s a -like i i has a pe ec ma ching Msuch ha GMis a s a .
De ini ion (3-Connec i i y)
A g aph Gis 3-connec ed i emo ing 2 e ices does no disconnec i .
Theo em (Main Resul )
E e y 3-connec ed s a -like g aph has a 3-decomposi ion.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 5 / 15

Ou Resul
De ini ion (S a -like G aphs)
Le Gbe a connec ed cubic g aph wi h a pe ec ma ching M.
IG−Mconsis s o cycles.
ICon ac ing hese yields he con ac ion g aph GM.
Gis s a -like i i has a pe ec ma ching Msuch ha GMis a s a .
De ini ion (3-Connec i i y)
A g aph Gis 3-connec ed i emo ing 2 e ices does no disconnec i .
Theo em (Main Resul )
E e y 3-connec ed s a -like g aph has a 3-decomposi ion.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 5 / 15
Ou Resul
De ini ion (S a -like G aphs)
Le Gbe a connec ed cubic g aph wi h a pe ec ma ching M.
IG−Mconsis s o cycles.
ICon ac ing hese yields he con ac ion g aph GM.
Gis s a -like i i has a pe ec ma ching Msuch ha GMis a s a .
De ini ion (3-Connec i i y)
A g aph Gis 3-connec ed i emo ing 2 e ices does no disconnec i .
Theo em (Main Resul )
E e y 3-connec ed s a -like g aph has a 3-decomposi ion.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 5 / 15
Ou Resul
De ini ion (S a -like G aphs)
Le Gbe a connec ed cubic g aph wi h a pe ec ma ching M.
IG−Mconsis s o cycles.
ICon ac ing hese yields he con ac ion g aph GM.
Gis s a -like i i has a pe ec ma ching Msuch ha GMis a s a .
De ini ion (3-Connec i i y)
A g aph Gis 3-connec ed i emo ing 2 e ices does no disconnec i .
Theo em (Main Resul )
E e y 3-connec ed s a -like g aph has a 3-decomposi ion.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 5 / 15
Ou Resul
De ini ion (S a -like G aphs)
Le Gbe a connec ed cubic g aph wi h a pe ec ma ching M.
IG−Mconsis s o cycles.
ICon ac ing hese yields he con ac ion g aph GM.
Gis s a -like i i has a pe ec ma ching Msuch ha GMis a s a .
De ini ion (3-Connec i i y)
A g aph Gis 3-connec ed i emo ing 2 e ices does no disconnec i .
Theo em (Main Resul )
E e y 3-connec ed s a -like g aph has a 3-decomposi ion.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 5 / 15
Ou Resul
De ini ion (S a -like G aphs)
Le Gbe a connec ed cubic g aph wi h a pe ec ma ching M.
IG−Mconsis s o cycles.
ICon ac ing hese yields he con ac ion g aph GM.
Gis s a -like i i has a pe ec ma ching Msuch ha GMis a s a .
De ini ion (3-Connec i i y)
A g aph Gis 3-connec ed i emo ing 2 e ices does no disconnec i .
Theo em (Main Resul )
E e y 3-connec ed s a -like g aph has a 3-decomposi ion.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 5 / 15

Ou Resul
De ini ion (S a -like G aphs)
Le Gbe a connec ed cubic g aph wi h a pe ec ma ching M.
IG−Mconsis s o cycles.
ICon ac ing hese yields he con ac ion g aph GM.
Gis s a -like i i has a pe ec ma ching Msuch ha GMis a s a .
De ini ion (3-Connec i i y)
A g aph Gis 3-connec ed i emo ing 2 e ices does no disconnec i .
Theo em (Main Resul )
E e y 3-connec ed s a -like g aph has a 3-decomposi ion.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 5 / 15
Ou Resul
De ini ion (S a -like G aphs)
Le Gbe a connec ed cubic g aph wi h a pe ec ma ching M.
IG−Mconsis s o cycles.
ICon ac ing hese yields he con ac ion g aph GM.
Gis s a -like i i has a pe ec ma ching Msuch ha GMis a s a .
De ini ion (3-Connec i i y)
A g aph Gis 3-connec ed i emo ing 2 e ices does no disconnec i .
Theo em (Xie, Zhou, Zhou)
E e y g aph wi h a con ac ion g aph on 3 e ices has a 3-decomposi ion.
Theo em (Main Resul )
E e y 3-connec ed s a -like g aph has a 3-decomposi ion.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 5 / 15
Ou Resul
De ini ion (S a -like G aphs)
Le Gbe a connec ed cubic g aph wi h a pe ec ma ching M.
IG−Mconsis s o cycles.
ICon ac ing hese yields he con ac ion g aph GM.
Gis s a -like i i has a pe ec ma ching Msuch ha GMis a s a .
De ini ion (3-Connec i i y)
A g aph Gis 3-connec ed i emo ing 2 e ices does no disconnec i .
Theo em (Main Resul )
E e y 3-connec ed s a -like g aph has a 3-decomposi ion.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 5 / 15
Hamil onian G aphs ha e 3-Decomposi ions
Theo em (Akba i, Jensen, Sigge s)
E e y connec ed cubic Hamil onian g aph has a 3-decomposi ion.
cho dless cycle
Call his he decomposi ion gi en by he cho dless cycle.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 6 / 15
Hamil onian G aphs ha e 3-Decomposi ions
Theo em (Akba i, Jensen, Sigge s)
E e y connec ed cubic Hamil onian g aph has a 3-decomposi ion.
cho dless cycle
Call his he decomposi ion gi en by he cho dless cycle.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 6 / 15

Hamil onian G aphs ha e 3-Decomposi ions
Theo em (Akba i, Jensen, Sigge s)
E e y connec ed cubic Hamil onian g aph has a 3-decomposi ion.
cho dless cycle
Call his he decomposi ion gi en by he cho dless cycle.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 6 / 15
Hamil onian G aphs ha e 3-Decomposi ions
Theo em (Akba i, Jensen, Sigge s)
E e y connec ed cubic Hamil onian g aph has a 3-decomposi ion.
cho dless cycle
Call his he decomposi ion gi en by he cho dless cycle.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 6 / 15
Hamil onian G aphs ha e 3-Decomposi ions
Theo em (Akba i, Jensen, Sigge s)
E e y connec ed cubic Hamil onian g aph has a 3-decomposi ion.
cho dless cycle
Call his he decomposi ion gi en by he cho dless cycle.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 6 / 15
Hamil onian G aphs ha e 3-Decomposi ions
Theo em (Akba i, Jensen, Sigge s)
E e y connec ed cubic Hamil onian g aph has a 3-decomposi ion.
cho dless cycle
Call his he decomposi ion gi en by he cho dless cycle.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 6 / 15
Fi s S eps
Theo em (Main Resul )
E e y 3-connec ed s a -like g aph has a 3-decomposi ion.
Assump ion
All cycles in G−Mha e cho ds in G.
Obse a ion
The e exis s a cho dless cycle a oiding any ixed e ex.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 7 / 15

Fi s S eps
Theo em (Main Resul )
E e y 3-connec ed s a -like g aph has a 3-decomposi ion.
Assump ion
All cycles in G−Mha e cho ds in G.
Obse a ion
The e exis s a cho dless cycle a oiding any ixed e ex.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 7 / 15
Fi s S eps
Theo em (Main Resul )
E e y 3-connec ed s a -like g aph has a 3-decomposi ion.
Assump ion
All cycles in G−Mha e cho ds in G.
Obse a ion
The e exis s a cho dless cycle a oiding any ixed e ex.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 7 / 15
Fi s S eps
Theo em (Main Resul )
E e y 3-connec ed s a -like g aph has a 3-decomposi ion.
Assump ion
All cycles in G−Mha e cho ds in G.
Obse a ion
The e exis s a cho dless cycle a oiding any ixed e ex.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 7 / 15
Fi s S eps
Theo em (Main Resul )
E e y 3-connec ed s a -like g aph has a 3-decomposi ion.
Assump ion
All cycles in G−Mha e cho ds in G.
Obse a ion
The e exis s a cho dless cycle a oiding any ixed e ex.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 7 / 15
Decomposing he Cen e
ITake any cho dless cycle C.
ICheck amoun o edges o
each ip.
IA mos 1.
IA leas 2 o some ip.
IUse decomposi ion
Igi en by C.
Iusing a pa o C.
I3 cases o he ips:
Ionly black edges,
I1 g een edge,
I2 ed edges.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 8 / 15

Decomposing he Cen e
ITake any cho dless cycle C.
ICheck amoun o edges o
each ip.
IA mos 1.
IA leas 2 o some ip.
IUse decomposi ion
Igi en by C.
Iusing a pa o C.
I3 cases o he ips:
Ionly black edges,
I1 g een edge,
I2 ed edges.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 8 / 15
Decomposing he Cen e
ITake any cho dless cycle C.
ICheck amoun o edges o
each ip.
IA mos 1.
IA leas 2 o some ip.
IUse decomposi ion
Igi en by C.
Iusing a pa o C.
I3 cases o he ips:
Ionly black edges,
I1 g een edge,
I2 ed edges.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 8 / 15
Decomposing he Cen e
ITake any cho dless cycle C.
ICheck amoun o edges o
each ip.
IA mos 1.
IA leas 2 o some ip.
IUse decomposi ion
Igi en by C.
Iusing a pa o C.
I3 cases o he ips:
Ionly black edges,
I1 g een edge,
I2 ed edges.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 8 / 15
Decomposing he Cen e
ITake any cho dless cycle C.
ICheck amoun o edges o
each ip.
IA mos 1.
IA leas 2 o some ip.
IUse decomposi ion
Igi en by C.
Iusing a pa o C.
I3 cases o he ips:
Ionly black edges,
I1 g een edge,
I2 ed edges.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 8 / 15
The Tips: Only Black Edges
Take he decomposi ion gi en by a cho dless cycle:
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 9 / 15

The Tips: Only Black Edges
Take he decomposi ion gi en by a cho dless cycle:
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 9 / 15
The Tips: Only Black Edges
Take he decomposi ion gi en by a cho dless cycle:
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 9 / 15
The Tips: One G een Edge
ILook o a use ul cho dless cycle.
II none exis s ⇒All cho ds go om he le o he igh pa h.
IBo h pa hs ha e he same leng h ⇒Le els.
ICall a cho d long i i s ends a e a leas 2 le els apa .
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 10 / 15
The Tips: One G een Edge
ILook o a use ul cho dless cycle.
II none exis s ⇒All cho ds go om he le o he igh pa h.
IBo h pa hs ha e he same leng h ⇒Le els.
ICall a cho d long i i s ends a e a leas 2 le els apa .
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 10 / 15
The Tips: One G een Edge
Case 1: No long cho d exis s
ITwo o m o cho ds:
Idi ec
Ic oss
IOb ain a Hamil onian pa h.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 11 / 15

The Tips: One G een Edge
Case 1: No long cho d exis s
ITwo o m o cho ds:
Idi ec
Ic oss
IOb ain a Hamil onian pa h.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 11 / 15
The Tips: One G een Edge
Case 1: No long cho d exis s
ITwo o m o cho ds:
Idi ec
Ic oss
IOb ain a Hamil onian pa h.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 11 / 15
The Tips: One G een Edge
Case 1: No long cho d exis s
ITwo o m o cho ds:
Idi ec
Ic oss
IOb ain a Hamil onian pa h.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 11 / 15
The Tips: One G een Edge
Case 1: No long cho d exis s
ITwo o m o cho ds:
Idi ec
Ic oss
IOb ain a Hamil onian pa h.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 11 / 15
The Tips: One G een Edge
Case 1: No long cho d exis s
ITwo o m o cho ds:
Idi ec
Ic oss
IOb ain a Hamil onian pa h.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 11 / 15

The Tips: One G een Edge
Case 2: A long cho d exis s
ITake he i s such cho d.
IVe ices be o e pai ed up.
IRega d he o he e ex on
his le el and i s successo .
INeighbou s below he cho d.
IUse a wo-cho d cycle.
ILea es a ee.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 12 / 15
The Tips: One G een Edge
Case 2: A long cho d exis s
ITake he i s such cho d.
IVe ices be o e pai ed up.
IRega d he o he e ex on
his le el and i s successo .
INeighbou s below he cho d.
IUse a wo-cho d cycle.
ILea es a ee.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 12 / 15
The Tips: One G een Edge
Case 2: A long cho d exis s
ITake he i s such cho d.
IVe ices be o e pai ed up.
IRega d he o he e ex on
his le el and i s successo .
INeighbou s below he cho d.
IUse a wo-cho d cycle.
ILea es a ee.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 12 / 15
The Tips: One G een Edge
Case 2: A long cho d exis s
ITake he i s such cho d.
IVe ices be o e pai ed up.
IRega d he o he e ex on
his le el and i s successo .
INeighbou s below he cho d.
IUse a wo-cho d cycle.
ILea es a ee.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 12 / 15
The Tips: Two Red Edges
IOne pa h connec s o cen e.
IPu i in Tand es in C.
IP oblem: “ ed” cho ds.
IIdea: use o sho cu .
IMus connec new pa hs.
ITake maximal sequence o
cho ds such ha
Ipa hs can be connec ed,
Icho ds a e maximal.
INeed: No mo e ed cho ds.
IBy maximali y: No cho ds
be ween di e en ed pa hs.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 13 / 15

The Tips: Two Red Edges
IOne pa h connec s o cen e.
IPu i in Tand es in C.
IP oblem: “ ed” cho ds.
IIdea: use o sho cu .
IMus connec new pa hs.
ITake maximal sequence o
cho ds such ha
Ipa hs can be connec ed,
Icho ds a e maximal.
INeed: No mo e ed cho ds.
IBy maximali y: No cho ds
be ween di e en ed pa hs.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 13 / 15
The Tips: Two Red Edges
IOne pa h connec s o cen e.
IPu i in Tand es in C.
IP oblem: “ ed” cho ds.
IIdea: use o sho cu .
IMus connec new pa hs.
ITake maximal sequence o
cho ds such ha
Ipa hs can be connec ed,
Icho ds a e maximal.
INeed: No mo e ed cho ds.
IBy maximali y: No cho ds
be ween di e en ed pa hs.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 13 / 15
The Tips: Two Red Edges
IOne pa h connec s o cen e.
IPu i in Tand es in C.
IP oblem: “ ed” cho ds.
IIdea: use o sho cu .
IMus connec new pa hs.
ITake maximal sequence o
cho ds such ha
Ipa hs can be connec ed,
Icho ds a e maximal.
INeed: No mo e ed cho ds.
IBy maximali y: No cho ds
be ween di e en ed pa hs.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 13 / 15
The Tips: Two Red Edges
IOne pa h connec s o cen e.
IPu i in Tand es in C.
IP oblem: “ ed” cho ds.
IIdea: use o sho cu .
IMus connec new pa hs.
ITake maximal sequence o
cho ds such ha
Ipa hs can be connec ed,
Icho ds a e maximal.
INeed: No mo e ed cho ds.
IBy maximali y: No cho ds
be ween di e en ed pa hs.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 13 / 15
The Tips: Two Red Edges
IOne pa h connec s o cen e.
IPu i in Tand es in C.
IP oblem: “ ed” cho ds.
IIdea: use o sho cu .
IMus connec new pa hs.
ITake maximal sequence o
cho ds such ha
Ipa hs can be connec ed,
Icho ds a e maximal.
INeed: No mo e ed cho ds.
IBy maximali y: No cho ds
be ween di e en ed pa hs.
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 13 / 15

The Tips: Two Red Edges
ICall e ices wi h neighbou s in
g een pa hs o he cen e good.
ISuppose no all e ices in a ed
pa h a e good.
ITake wo good e ices
Iwi h a bad one in be ween,
Io minimal dis ance.
IThe e exis s a “c ossing” cho d.
IEx ends he sequence. E
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 14 / 15
The Tips: Two Red Edges
ICall e ices wi h neighbou s in
g een pa hs o he cen e good.
ISuppose no all e ices in a ed
pa h a e good.
ITake wo good e ices
Iwi h a bad one in be ween,
Io minimal dis ance.
IThe e exis s a “c ossing” cho d.
IEx ends he sequence. E
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 14 / 15
The Tips: Two Red Edges
ICall e ices wi h neighbou s in
g een pa hs o he cen e good.
ISuppose no all e ices in a ed
pa h a e good.
ITake wo good e ices
Iwi h a bad one in be ween,
Io minimal dis ance.
IThe e exis s a “c ossing” cho d.
IEx ends he sequence. E
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 14 / 15
The Tips: Two Red Edges
ICall e ices wi h neighbou s in
g een pa hs o he cen e good.
ISuppose no all e ices in a ed
pa h a e good.
ITake wo good e ices
Iwi h a bad one in be ween,
Io minimal dis ance.
IThe e exis s a “c ossing” cho d.
IEx ends he sequence. E
Oli e Bach le , S en O. K umke Decomposing S a -like Cubic G aphs YMSGR 2020 14 / 15