resolution of the oberwolfach problem · resolution the oberwolfach problem has a solution for all...
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Resolution of the Oberwolfach problem
Daniela Kuhn
joint work with Stefan Glock, Felix Joos, Jaehoon Kim and Deryk Osthus
University of Birmingham
April 2019
Daniela Kuhn Oberwolfach problem
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Decompositions
DefinitionAn F -decomposition of a graph G is a partition of the edge set ofG where each part is isomorphic to F .
•If G = Kn and F = K3, this is a Steiner triple system of order n
•Kirkman’s schoolgirl problem (1850): Does K15 decompose intotriangle factors?•Walecki’s theorem (1892): Kn has a decomposition intoHamilton cycles for every odd n
Common generalization: Oberwolfach problem
Daniela Kuhn Oberwolfach problem
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Formal statement
cycle factor = vertex disjoint cycles spanning all vertices
Oberwolfach problem (Ringel, 1967)Let F be any cycle factor on n vertices. Does Kn have anF -decomposition?
posed at Oberwolfach conference and can be rephrased as:
Oberwolfach problem (Ringel, 1967)Given round tables with n seats in total and n people who eat n−1
2meals together, is it possible to find a seating chart such thateveryone sits next to everyone else exactly once?
Daniela Kuhn Oberwolfach problem
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Formal statement
Oberwolfach problem (Ringel, 1967)Let F be any cycle factor on n vertices. Does Kn have anF -decomposition?
Example: F = C3 ·∪C4 ·∪C4
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Formal statement
Oberwolfach problem (Ringel, 1967)Let F be any cycle factor on n vertices. Does Kn have anF -decomposition?
Example: F = C3 ·∪C4 ·∪C4
123
4
5
6
78 9
10
11
Daniela Kuhn Oberwolfach problem
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Formal statement
Oberwolfach problem (Ringel, 1967)Let F be any cycle factor on n vertices. Does Kn have anF -decomposition?
Example: F = C3 ·∪C4 ·∪C4
123
4
5
6
78 9
10
11
Daniela Kuhn Oberwolfach problem
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Formal statement
Oberwolfach problem (Ringel, 1967)Let F be any cycle factor on n vertices. Does Kn have anF -decomposition?
Example: F = C3 ·∪C4 ·∪C4
123
4
5
6
78 9
10
11
Daniela Kuhn Oberwolfach problem
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Partial results
•F = Hamilton cycle: Walecki (1892)•F = triangle factor: Ray-Chaudhuri & Wilson, and Lu (1970s)...Theorem (Bryant and Scharaschkin, 2009)∃ infinitely many n such that for any cycle factor F on n vertices,Kn has F -decomposition....•Traetta (2013): solution if F consists of two cycles only•approximate versions by Ferber–Lee–Mousset andKim–Kuhn–Osthus–Tyomkyn (2017)...
≥ 100 research papers covering many partial results
Daniela Kuhn Oberwolfach problem
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Resolution
The Oberwolfach problem has a solution for all sufficiently large n.
Theorem (Glock, Joos, Kim, Kuhn, Osthus, 18+)∃n0 such that for all odd n ≥ n0 and any cycle factor F on nvertices, Kn has an F -decomposition.
•for even n, one can ask for a decompositionof Kn−perfect matching•Hamilton-Waterloo problem: two cycle factors F1,F2 given,and prescribed how often each of them is to be used in thedecomposition
We also solve these problems (for sufficiently large n).
Daniela Kuhn Oberwolfach problem
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Resolution
The Oberwolfach problem has a solution for all sufficiently large n.
Theorem (Glock, Joos, Kim, Kuhn, Osthus, 18+)∃n0 such that for all odd n ≥ n0 and any cycle factor F on nvertices, Kn has an F -decomposition.
•for even n, one can ask for a decompositionof Kn−perfect matching•Hamilton-Waterloo problem: two cycle factors F1,F2 given,and prescribed how often each of them is to be used in thedecomposition
We also solve these problems (for sufficiently large n).
Daniela Kuhn Oberwolfach problem
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Most general statement:
TheoremSuppose 1/n� ξ� 1/∆,α < 1. Let G be an r-regular n-vertexgraph with r ≥ (1− ξ)n and let F ,H be collections of graphssatisfying the following:
F is a collection of at least αn copies of F , where F is a2-regular n-vertex graph;each H ∈H is a ξ-separable n-vertex rH -regular graph forsome rH ≤∆;e(F ∪H) = e(G).
Then G decomposes into F ∪H.
⇒ can choose first ξn factors greedily‘Separable’=‘small bandwidth’=2-factors, powers of cycles,H-factors...
Daniela Kuhn Oberwolfach problem
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Proof sketch: simplified setup
A C`-decomposition of G is resolvable if it can be partitioned intoC`-factors.So if F is a C`-factor, then an F -decomposition is precisely aresolvable C`-decomposition.(existence of resolvable C`-decompositions in Kn proved byAlspach, Schellenberg, Stinson, Wagner)
But F might consist of cycles of arbitrary lengths.Approach: reduce the problem of finding F -decomposition tofinding resolvable C`-decompositions in a quasi-random graphs, for` ∈ {3,4,5}.
Daniela Kuhn Oberwolfach problem
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Proof sketch: simplified setup
A C`-decomposition of G is resolvable if it can be partitioned intoC`-factors.So if F is a C`-factor, then an F -decomposition is precisely aresolvable C`-decomposition.(existence of resolvable C`-decompositions in Kn proved byAlspach, Schellenberg, Stinson, Wagner)
But F might consist of cycles of arbitrary lengths.Approach: reduce the problem of finding F -decomposition tofinding resolvable C`-decompositions in a quasi-random graphs, for` ∈ {3,4,5}.
Daniela Kuhn Oberwolfach problem
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Rewiring: simplified setup
Suppose all cycle lengths `1, . . . , `t in F are divisible by 3, and weseek F -decomposition of Kn,n,n, where n =∑
`i/3.
V3
V1
V2
ππ(w)
wv z
w
(w ,z) ∈ π
Let π be a permutation on V3 with cycles of lengths `1/3, . . . , `t/3.Given H ⊆ Kn,n,n, obtain π(H) by replacing vw with vπ(w)whenever v ∈ V2,w ∈ V3
H = C3-factor ⇒ π(H) = F
resolvable C3-decomposition ⇒ F -decomposition(resolvable C3-decomposition if Kn,n,n exists, eg by Bose,Shrikhande and Parker)
Daniela Kuhn Oberwolfach problem
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Rewiring: simplified setup
Suppose all cycle lengths `1, . . . , `t in F are divisible by 3, and weseek F -decomposition of Kn,n,n, where n =∑
`i/3.
V3
V1
V2
π
π(w)
wv z
w
(w ,z) ∈ π
Let π be a permutation on V3 with cycles of lengths `1/3, . . . , `t/3.
Given H ⊆ Kn,n,n, obtain π(H) by replacing vw with vπ(w)whenever v ∈ V2,w ∈ V3
H = C3-factor ⇒ π(H) = F
resolvable C3-decomposition ⇒ F -decomposition(resolvable C3-decomposition if Kn,n,n exists, eg by Bose,Shrikhande and Parker)
Daniela Kuhn Oberwolfach problem
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Rewiring: simplified setup
Suppose all cycle lengths `1, . . . , `t in F are divisible by 3, and weseek F -decomposition of Kn,n,n, where n =∑
`i/3.
V3
V1
V2
ππ(w)
wv
zw
(w ,z) ∈ π
Let π be a permutation on V3 with cycles of lengths `1/3, . . . , `t/3.Given H ⊆ Kn,n,n, obtain π(H) by replacing vw with vπ(w)whenever v ∈ V2,w ∈ V3
H = C3-factor ⇒ π(H) = F
resolvable C3-decomposition ⇒ F -decomposition(resolvable C3-decomposition if Kn,n,n exists, eg by Bose,Shrikhande and Parker)
Daniela Kuhn Oberwolfach problem
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Rewiring: simplified setup
Suppose all cycle lengths `1, . . . , `t in F are divisible by 3, and weseek F -decomposition of Kn,n,n, where n =∑
`i/3.
V3
V1
V2
ππ(w)
wv z
w
(w ,z) ∈ π
Let π be a permutation on V3 with cycles of lengths `1/3, . . . , `t/3.Given H ⊆ Kn,n,n, obtain π(H) by replacing vw with vπ(w)whenever v ∈ V2,w ∈ V3
H = C3-factor ⇒ π(H) = F
resolvable C3-decomposition ⇒ F -decomposition(resolvable C3-decomposition if Kn,n,n exists, eg by Bose,Shrikhande and Parker)
Daniela Kuhn Oberwolfach problem
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Rewiring: simplified setup
Suppose all cycle lengths `1, . . . , `t in F are divisible by 3, and weseek F -decomposition of Kn,n,n, where n =∑
`i/3.
V3
V1
V2
ππ(w)
wv z
w
(w ,z) ∈ π
Let π be a permutation on V3 with cycles of lengths `1/3, . . . , `t/3.Given H ⊆ Kn,n,n, obtain π(H) by replacing vw with vπ(w)whenever v ∈ V2,w ∈ V3
H = C3-factor ⇒ π(H) = F
resolvable C3-decomposition ⇒ F -decomposition(resolvable C3-decomposition if Kn,n,n exists, eg by Bose,Shrikhande and Parker)
Daniela Kuhn Oberwolfach problem
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Rewiring: simplified setup
Suppose all cycle lengths `1, . . . , `t in F are divisible by 3, and weseek F -decomposition of Kn,n,n, where n =∑
`i/3.
V3
V1
V2
ππ(w)
wv z
w
(w ,z) ∈ π
Let π be a permutation on V3 with cycles of lengths `1/3, . . . , `t/3.Given H ⊆ Kn,n,n, obtain π(H) by replacing vw with vπ(w)whenever v ∈ V2,w ∈ V3
H = C3-factor ⇒ π(H) = F
resolvable C3-decomposition ⇒ F -decomposition(resolvable C3-decomposition if Kn,n,n exists, eg by Bose,Shrikhande and Parker)
Daniela Kuhn Oberwolfach problem
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Rewiring: simplified setup
Suppose all cycle lengths `1, . . . , `t in F are divisible by 3, and weseek F -decomposition of Kn,n,n, where n =∑
`i/3.
V3
V1
V2
ππ(w)
wv z
w
(w ,z) ∈ π
Let π be a permutation on V3 with cycles of lengths `1/3, . . . , `t/3.Given H ⊆ Kn,n,n, obtain π(H) by replacing vw with vπ(w)whenever v ∈ V2,w ∈ V3
H = C3-factor ⇒ π(H) = Fresolvable C3-decomposition ⇒ F -decomposition(resolvable C3-decomposition if Kn,n,n exists, eg by Bose,Shrikhande and Parker)
Daniela Kuhn Oberwolfach problem
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Proof sketch: general setup
Suppose aim to find F -decomposition of Kn.
Absorption approach1 Take out highly structured absorbing subgraph A.2 Find approximate decomposition of Kn−A into copies of F to
leave a sparse leftover L.3 Use ‘structure’ of A to find F -decomposition of A∪L?
First used in context of decompositions for proof of Kelly’sconjecture on Hamilton decompositions (Kuhn, Osthus’13)
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Tool for approximate decomposition
Can apply special case of ‘bandwidth theorem for approximatedecompositions’:
Theorem (Condon, Kim, Kuhn, Osthus, 2017+)Suppose H1, . . . ,Hs is a collection of n-vertex 2-regular graphs andG is an n-vertex d-regular graph such that
d ≥ (1 + o(1))9n/10 and s ≤ (1−o(1))d/2.
Then H1, . . . ,Hs pack into G.
Proof is based on:blow-up lemma for approximate decompositionsSzemeredi’s regularity lemma
actually prove version for general bounded degree graphs of smallbandwidth
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The absorbing structure
The absorbing structure A:grey/black graphs between classes are quasi-random
π
Can show that A has an F -decomposition via a generalization ofswitching permutation argument described earlier(by reducing to resolvable C3, C4 and C5-decompositions)
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Tool for finding resolvable decomposition
To show absorber has an F -decomposition (via switchingpermutation argument), we use:Resolvable C`-decompositions exist in quasirandom partite graphs
Theorem (Keevash, 2018+)Suppose that G is a blow-up of an `-cycle so that each blown-uppair is quasirandom and regular. Then G has a resolvableC`-decomposition.
Note this follows from:the existence of `-wheel decompositions in a quasi-random blow-upof an `-wheel
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The absorbing structure
The absorbing structure A:
π
Let L be the leftover from the approximate decomposition step.(1) Use suitable edges E of A to cover L with copies of F(2) Then decompose A−E into copies of F ,
(by reducing to resolvable C3, C4, and C5-decomposition).
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Proof sketch: general setupRecall: L is the leftover from the approximate decomposition step.
(1) Use suitable edges E of A to cover L with copies of F(2) Then decompose A−E into copies of F ,
(by reducing to resolvable C3, C4, and C5-decomposition).For (1), decompose L into small matchings Mi and extend each Miinto a copy Mi ∪Ei of F using edges Ei of A
Challenges/Problems:Need to augment A by adding edges inside clusters in order tocover edges of L between clustersNeed to do the extension in a ‘globally balanced’ way,i.e. E = ∪i Ei is ‘balanced’ with respect to A.This ensures that A−E is still F -decomposable.Above approach only works if there are ηn vertices of F inlong cycles
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Proof sketch: general setupRecall: L is the leftover from the approximate decomposition step.
(1) Use suitable edges E of A to cover L with copies of F(2) Then decompose A−E into copies of F ,
(by reducing to resolvable C3, C4, and C5-decomposition).For (1), decompose L into small matchings Mi and extend each Miinto a copy Mi ∪Ei of F using edges Ei of A
Challenges/Problems:Need to augment A by adding edges inside clusters in order tocover edges of L between clustersNeed to do the extension in a ‘globally balanced’ way,i.e. E = ∪i Ei is ‘balanced’ with respect to A.This ensures that A−E is still F -decomposable.Above approach only works if there are ηn vertices of F inlong cycles
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Open problems and related questions
Not only 2-regular graphs?
ConjectureSuppose ∆� n. Let F1, . . . ,Ft be n-vertex graphs such that Fi isri -regular for some ri ≤∆ and
∑i∈[t] ri = n−1. Then there is a
decomposition of Kn into F1, . . . ,Ft .
Interesting special caseFi is a kth power of a Hamilton cycle
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Open problems: Hamilton decompositions of hypergraphs
Theorem (Walecki, 1892)Complete graph Kn has a Hamilton decomposition ⇔ n odd
Problem: Prove a hypergraph version of Walecki’s theorem.
loose Hamilton cycle tight Hamilton cycle Berge Hamilton cycle
open open solved (K+O)
approximate versions exist for the loose and tight case(Bal, Frieze, Krivelevich, Loh) for infinitely many n
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Open problems: Euler circuits
Conjecture (Chung, Diaconis and Graham, 1989 ($100))For sufficiently large n, K k
n has a tight Euler tour iff k |(n−1
k−1).
Curtis, Hines, Hurlbert, Moyer (2009): approximate solution
CHHM: At the 2004 Banff Workshop ... it was suggested.. that amodest inflationary rate should revalue the prize near 250.04....Due to our proof that near-universal cycles exist, we believe thatwe deserve asymptotically much of the prize money, or(1−o(1))(250.04). Since we do not know the speed of the o(1)term, we have made a conservative estimate of 249.99.
Theorem (Glock, Joos, Kuhn, Osthus, 18+)The conjecture is true.
based on existence of F -designs (Glock, Lo, Kuhn, Osthus, 17+)
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Open problems: Euler circuits
Conjecture (Chung, Diaconis and Graham, 1989 ($100))For sufficiently large n, K k
n has a tight Euler tour iff k |(n−1
k−1).
Curtis, Hines, Hurlbert, Moyer (2009): approximate solutionCHHM: At the 2004 Banff Workshop ... it was suggested.. that amodest inflationary rate should revalue the prize near 250.04....
Due to our proof that near-universal cycles exist, we believe thatwe deserve asymptotically much of the prize money, or(1−o(1))(250.04). Since we do not know the speed of the o(1)term, we have made a conservative estimate of 249.99.
Theorem (Glock, Joos, Kuhn, Osthus, 18+)The conjecture is true.
based on existence of F -designs (Glock, Lo, Kuhn, Osthus, 17+)
Daniela Kuhn Oberwolfach problem
![Page 32: Resolution of the Oberwolfach problem · Resolution The Oberwolfach problem has a solution for all sufficiently large n. Theorem (Glock, Joos, Kim, K¨uhn, Osthus, 18 +) ∃n 0 such](https://reader034.vdocuments.us/reader034/viewer/2022050716/5e321e91cbeae64ff23a8228/html5/thumbnails/32.jpg)
Open problems: Euler circuits
Conjecture (Chung, Diaconis and Graham, 1989 ($100))For sufficiently large n, K k
n has a tight Euler tour iff k |(n−1
k−1).
Curtis, Hines, Hurlbert, Moyer (2009): approximate solutionCHHM: At the 2004 Banff Workshop ... it was suggested.. that amodest inflationary rate should revalue the prize near 250.04....Due to our proof that near-universal cycles exist, we believe thatwe deserve asymptotically much of the prize money, or(1−o(1))(250.04). Since we do not know the speed of the o(1)term, we have made a conservative estimate of 249.99.
Theorem (Glock, Joos, Kuhn, Osthus, 18+)The conjecture is true.
based on existence of F -designs (Glock, Lo, Kuhn, Osthus, 17+)
Daniela Kuhn Oberwolfach problem
![Page 33: Resolution of the Oberwolfach problem · Resolution The Oberwolfach problem has a solution for all sufficiently large n. Theorem (Glock, Joos, Kim, K¨uhn, Osthus, 18 +) ∃n 0 such](https://reader034.vdocuments.us/reader034/viewer/2022050716/5e321e91cbeae64ff23a8228/html5/thumbnails/33.jpg)
Open problems: Euler circuits
Conjecture (Chung, Diaconis and Graham, 1989 ($100))For sufficiently large n, K k
n has a tight Euler tour iff k |(n−1
k−1).
Curtis, Hines, Hurlbert, Moyer (2009): approximate solutionCHHM: At the 2004 Banff Workshop ... it was suggested.. that amodest inflationary rate should revalue the prize near 250.04....Due to our proof that near-universal cycles exist, we believe thatwe deserve asymptotically much of the prize money, or(1−o(1))(250.04). Since we do not know the speed of the o(1)term, we have made a conservative estimate of 249.99.
Theorem (Glock, Joos, Kuhn, Osthus, 18+)The conjecture is true.
based on existence of F -designs (Glock, Lo, Kuhn, Osthus, 17+)
Daniela Kuhn Oberwolfach problem
![Page 34: Resolution of the Oberwolfach problem · Resolution The Oberwolfach problem has a solution for all sufficiently large n. Theorem (Glock, Joos, Kim, K¨uhn, Osthus, 18 +) ∃n 0 such](https://reader034.vdocuments.us/reader034/viewer/2022050716/5e321e91cbeae64ff23a8228/html5/thumbnails/34.jpg)
Open problems: Euler circuits
Theorem (Glock, Joos, Kuhn, Osthus, 18+)For sufficiently large n, K k
n has a tight Euler tour if and only ifk |(n−1
k−1).
ConjectureEvery k-graph G with δk−1(G)≥ (1/2 + o(1))n has a tight Eulertour if all vertex degrees are divisible by k.
Daniela Kuhn Oberwolfach problem
![Page 35: Resolution of the Oberwolfach problem · Resolution The Oberwolfach problem has a solution for all sufficiently large n. Theorem (Glock, Joos, Kim, K¨uhn, Osthus, 18 +) ∃n 0 such](https://reader034.vdocuments.us/reader034/viewer/2022050716/5e321e91cbeae64ff23a8228/html5/thumbnails/35.jpg)
Bon appetit!
Daniela Kuhn Oberwolfach problem