web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small...
Transcript of web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small...
![Page 1: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/1.jpg)
Decompositions of dense graphs into smallsubgraphs
Deryk Osthus
joint work with Ben Barber, Daniela Kuhn, Allan Lo
University of Birmingham
August 2015
![Page 2: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/2.jpg)
F -decompositions
Definition
A graph G has an F -decomposition if the edges of G can be coveredby edge-disjoint copies of F .
Question
When does G have an F -decomposition?
![Page 3: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/3.jpg)
F -decompositions
Definition
A graph G has an F -decomposition if the edges of G can be coveredby edge-disjoint copies of F .
Question
When does G have an F -decomposition?
![Page 4: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/4.jpg)
F -decompositions
Definition
A graph G has an F -decomposition if the edges of G can be coveredby edge-disjoint copies of F .
Question
When does G have an F -decomposition?
![Page 5: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/5.jpg)
Necessary conditions
Question
When does G have an F -decomposition?
If G has a triangle decomposition, then
(a) the number of edges of G is divisible by 3;
(b) every vertex has even degree.
A necessary condition
If G has an F -decomposition, then
(a) the number of edges in F divides the number of edges in G;
(b) gcd(F) divides gcd(G), where gcd(H) is the largest integerdividing the degree of every vertex of a graph H.
G is said to be F -divisible if G satisfies (a) and (b).
![Page 6: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/6.jpg)
Necessary conditions
Question
When does G have an F -decomposition?
If G has a triangle decomposition, then
(a) the number of edges of G is divisible by 3;
(b) every vertex has even degree.
A necessary condition
If G has an F -decomposition, then
(a) the number of edges in F divides the number of edges in G;
(b) gcd(F) divides gcd(G), where gcd(H) is the largest integerdividing the degree of every vertex of a graph H.
G is said to be F -divisible if G satisfies (a) and (b).
![Page 7: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/7.jpg)
F -divisibility
F -divisibility
(a) the number of edges in F divides the number of edges in G
(b) gcd(F) divides gcd(G), where gcd(H) is the largest integerdividing the degree of every vertex of a graph H
F -divisiblity is not sufficient for F -decomposition.
The problem of deciding whether a graph G has an F -decompositionis NP-complete if F contains a connected component with at least 3edges.
![Page 8: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/8.jpg)
F -divisibility
F -divisibility
(a) the number of edges in F divides the number of edges in G
(b) gcd(F) divides gcd(G), where gcd(H) is the largest integerdividing the degree of every vertex of a graph H
F -divisiblity is not sufficient for F -decomposition.
The problem of deciding whether a graph G has an F -decompositionis NP-complete if F contains a connected component with at least 3edges.
![Page 9: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/9.jpg)
Decompositions of complete host graphs
Theorem (Kirkman 1847)
Every triangle-divisible Kn has a triangle decomposition. (i.e. n ≡ 1,3mod 6)
Theorem (Wilson 1975)
For n large, every F -divisible Kn has an F -decomposition.
Generalization to hypergraph cliques:
Theorem (Keevash 2014+)
For r ≤ q� n, every complete r -uniform hypergraph on n verticesK (r)
n (subject to the necessary divisibility conditions) has aK (r)
q -decomposition.
![Page 10: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/10.jpg)
Decompositions of complete host graphs
Theorem (Kirkman 1847)
Every triangle-divisible Kn has a triangle decomposition. (i.e. n ≡ 1,3mod 6)
Theorem (Wilson 1975)
For n large, every F -divisible Kn has an F -decomposition.
Generalization to hypergraph cliques:
Theorem (Keevash 2014+)
For r ≤ q� n, every complete r -uniform hypergraph on n verticesK (r)
n (subject to the necessary divisibility conditions) has aK (r)
q -decomposition.
![Page 11: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/11.jpg)
Decompositions of complete host graphs
Theorem (Kirkman 1847)
Every triangle-divisible Kn has a triangle decomposition. (i.e. n ≡ 1,3mod 6)
Theorem (Wilson 1975)
For n large, every F -divisible Kn has an F -decomposition.
Generalization to hypergraph cliques:
Theorem (Keevash 2014+)
For r ≤ q� n, every complete r -uniform hypergraph on n verticesK (r)
n (subject to the necessary divisibility conditions) has aK (r)
q -decomposition.
![Page 12: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/12.jpg)
Decompositions of complete host graphs
Theorem (Kirkman 1847)
Every triangle-divisible Kn has a triangle decomposition. (i.e. n ≡ 1,3mod 6)
Theorem (Wilson 1975)
For n large, every F -divisible Kn has an F -decomposition.
Generalization to hypergraph cliques:
Theorem (Keevash 2014+)
For r ≤ q� n, every complete r -uniform hypergraph on n verticesK (r)
n (subject to the necessary divisibility conditions) has aK (r)
q -decomposition.
![Page 13: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/13.jpg)
Triangle decompositions of graphs of large minimum degree
Conjecture (Nash-Williams 1970)
Every large triangle-divisible graph G on n vertices with δ (G)≥ 3n/4has a triangle decomposition.
conjecture generalizes to Kr -decompositions
Extremal example: blow up each vertex of C4 to a Km (m odd anddivisible by 3).
Each triangle has at least one edge in one of the four cliques but lessthan a third of the edges lie inside the cliques.
![Page 14: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/14.jpg)
Triangle decompositions of graphs of large minimum degree
Conjecture (Nash-Williams 1970)
Every large triangle-divisible graph G on n vertices with δ (G)≥ 3n/4has a triangle decomposition.
conjecture generalizes to Kr -decompositionsExtremal example: blow up each vertex of C4 to a Km (m odd anddivisible by 3).
Each triangle has at least one edge in one of the four cliques but lessthan a third of the edges lie inside the cliques.
![Page 15: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/15.jpg)
Decompositions of graphs of large minimum degree
Theorem (Gustavsson 1991, Keevash 2014+)
For every graph F , there exist ε and n0 such that every F -divisiblegraph G on n ≥ n0 vertices with δ (G)≥ (1− ε)n has anF -decomposition.
For F = Kr , Gustavsson states ε = 10−37r−94.
Theorem (Yuster 2002)
Let F be a bipartite graph with δ (F) = 1. If G is F -divisible andδ (G)≥
(12 +o(1)
)n, then G has an F -decomposition.
Theorem (Bryant and Cavenagh 2014+)
If G is C4-divisible and δ (G)≥(
3132 +o(1)
)n, then G has a
C4-decomposition.
![Page 16: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/16.jpg)
Decompositions of graphs of large minimum degree
Theorem (Gustavsson 1991, Keevash 2014+)
For every graph F , there exist ε and n0 such that every F -divisiblegraph G on n ≥ n0 vertices with δ (G)≥ (1− ε)n has anF -decomposition.
For F = Kr , Gustavsson states ε = 10−37r−94.
Theorem (Yuster 2002)
Let F be a bipartite graph with δ (F) = 1. If G is F -divisible andδ (G)≥
(12 +o(1)
)n, then G has an F -decomposition.
Theorem (Bryant and Cavenagh 2014+)
If G is C4-divisible and δ (G)≥(
3132 +o(1)
)n, then G has a
C4-decomposition.
![Page 17: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/17.jpg)
Decompositions of graphs of large minimum degree
Theorem (Gustavsson 1991, Keevash 2014+)
For every graph F , there exist ε and n0 such that every F -divisiblegraph G on n ≥ n0 vertices with δ (G)≥ (1− ε)n has anF -decomposition.
For F = Kr , Gustavsson states ε = 10−37r−94.
Theorem (Yuster 2002)
Let F be a bipartite graph with δ (F) = 1. If G is F -divisible andδ (G)≥
(12 +o(1)
)n, then G has an F -decomposition.
Theorem (Bryant and Cavenagh 2014+)
If G is C4-divisible and δ (G)≥(
3132 +o(1)
)n, then G has a
C4-decomposition.
![Page 18: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/18.jpg)
Fractional decompositions
fractional F -decomposition of G: give every copy of F in G a weightw(F) ∈ [0,1] such that ∑F :e∈E(F) w(F) = 1 for each edge e of G
12
12
12
12
⇒ +
fractional decomposition threshold δfrac(F): smallest c ∈ [0,1] s.t.every large graph G with δ (G)≥ cn has fractional F -decomposition
fractional K3-decomposition threshold:Garaschuk (2014): δfrac(K3)≤ 0.956Dross (2015+): δfrac(K3)≤ 0.9
fractional Kr -decomposition threshold:Yuster (2005): δfrac(Kr )≤ 1− 1
9r10
Dukes (2012): δfrac(Kr )≤ 1− 116r4
Barber, Kuhn, Lo, Montgomery, Osthus (2015+): δfrac(Kr )≤ 1− 1104r3/2
![Page 19: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/19.jpg)
Fractional decompositions
fractional F -decomposition of G: give every copy of F in G a weightw(F) ∈ [0,1] such that ∑F :e∈E(F) w(F) = 1 for each edge e of G
12
12
12
12
⇒ +
fractional decomposition threshold δfrac(F): smallest c ∈ [0,1] s.t.every large graph G with δ (G)≥ cn has fractional F -decomposition
fractional K3-decomposition threshold:Garaschuk (2014): δfrac(K3)≤ 0.956Dross (2015+): δfrac(K3)≤ 0.9
fractional Kr -decomposition threshold:Yuster (2005): δfrac(Kr )≤ 1− 1
9r10
Dukes (2012): δfrac(Kr )≤ 1− 116r4
Barber, Kuhn, Lo, Montgomery, Osthus (2015+): δfrac(Kr )≤ 1− 1104r3/2
![Page 20: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/20.jpg)
Fractional decompositions
fractional F -decomposition of G: give every copy of F in G a weightw(F) ∈ [0,1] such that ∑F :e∈E(F) w(F) = 1 for each edge e of G
12
12
12
12
⇒ +
fractional decomposition threshold δfrac(F): smallest c ∈ [0,1] s.t.every large graph G with δ (G)≥ cn has fractional F -decomposition
fractional K3-decomposition threshold:Garaschuk (2014): δfrac(K3)≤ 0.956Dross (2015+): δfrac(K3)≤ 0.9
fractional Kr -decomposition threshold:Yuster (2005): δfrac(Kr )≤ 1− 1
9r10
Dukes (2012): δfrac(Kr )≤ 1− 116r4
Barber, Kuhn, Lo, Montgomery, Osthus (2015+): δfrac(Kr )≤ 1− 1104r3/2
![Page 21: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/21.jpg)
From fractional to ‘real’ decompositions
Theorem (Barber, Kuhn, Lo, Osthus 2014+)
Every large triangle-divisible graph G with δ (G)≥ (δfrac(K3)+o(1))nhas a triangle decomposition.
Theorem (Barber, Kuhn, Lo, Osthus 2014+)
If F is r -regular, then every large F -divisible graph G withδ (G)≥ (max{δfrac(F),1−1/3r}+o(1))n has a F -decomposition.
Corollary
Every large triangle-divisible graph G with δ (G)≥ (0.9+o(1))nhas a triangle decomposition.
Every large Kr -divisible graph G with δ (G)≥ (1−1/104r3/2)nhas a Kr -decomposition.
Every large F -divisible graph G with δ (G)≥ (1− c/|F |2)n hasan F -decomposition.
![Page 22: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/22.jpg)
From fractional to ‘real’ decompositions
Theorem (Barber, Kuhn, Lo, Osthus 2014+)
Every large triangle-divisible graph G with δ (G)≥ (δfrac(K3)+o(1))nhas a triangle decomposition.
Theorem (Barber, Kuhn, Lo, Osthus 2014+)
If F is r -regular, then every large F -divisible graph G withδ (G)≥ (max{δfrac(F),1−1/3r}+o(1))n has a F -decomposition.
Corollary
Every large triangle-divisible graph G with δ (G)≥ (0.9+o(1))nhas a triangle decomposition.
Every large Kr -divisible graph G with δ (G)≥ (1−1/104r3/2)nhas a Kr -decomposition.
Every large F -divisible graph G with δ (G)≥ (1− c/|F |2)n hasan F -decomposition.
![Page 23: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/23.jpg)
From fractional to ‘real’ decompositions
Theorem (Barber, Kuhn, Lo, Osthus 2014+)
Every large triangle-divisible graph G with δ (G)≥ (δfrac(K3)+o(1))nhas a triangle decomposition.
Theorem (Barber, Kuhn, Lo, Osthus 2014+)
If F is r -regular, then every large F -divisible graph G withδ (G)≥ (max{δfrac(F),1−1/3r}+o(1))n has a F -decomposition.
Corollary
Every large triangle-divisible graph G with δ (G)≥ (0.9+o(1))nhas a triangle decomposition.
Every large Kr -divisible graph G with δ (G)≥ (1−1/104r3/2)nhas a Kr -decomposition.
Every large F -divisible graph G with δ (G)≥ (1− c/|F |2)n hasan F -decomposition.
![Page 24: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/24.jpg)
From fractional to ‘real’ decompositions
Theorem (Barber, Kuhn, Lo, Osthus 2014+)
Every large triangle-divisible graph G with δ (G)≥ (δfrac(K3)+o(1))nhas a triangle decomposition.
Theorem (Barber, Kuhn, Lo, Osthus 2014+)
If F is r -regular, then every large F -divisible graph G withδ (G)≥ (max{δfrac(F),1−1/3r}+o(1))n has a F -decomposition.
Corollary
Every large triangle-divisible graph G with δ (G)≥ (0.9+o(1))nhas a triangle decomposition.
Every large Kr -divisible graph G with δ (G)≥ (1−1/104r3/2)nhas a Kr -decomposition.
Every large F -divisible graph G with δ (G)≥ (1− c/|F |2)n hasan F -decomposition.
![Page 25: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/25.jpg)
Proof idea
Theorem (Barber, Kuhn, Lo, Osthus 2014+)
Every large triangle-divisible graph G with δ (G)≥ (δfrac(K3)+o(1))nhas a triangle decomposition.
Will use:
Theorem (Haxell and Rodl 2001)
Every large graph G with δ (G)≥ δfrac(K3)n can be decomposed intoedge-disjoint copies of K3 and a remainder R with εn2 uncoverededges.
Problem: What to do with leftover?
![Page 26: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/26.jpg)
Proof idea
Theorem (Barber, Kuhn, Lo, Osthus 2014+)
Every large triangle-divisible graph G with δ (G)≥ (δfrac(K3)+o(1))nhas a triangle decomposition.
Will use:
Theorem (Haxell and Rodl 2001)
Every large graph G with δ (G)≥ δfrac(K3)n can be decomposed intoedge-disjoint copies of K3 and a remainder R with εn2 uncoverededges.
Problem: What to do with leftover?
![Page 27: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/27.jpg)
Planning ahead
Plan1 Take out highly structured subgraph A.2 Take out many triangles to leave a sparse remainder R.3 Use ‘structure’ of A to decompose A∪R into triangles.
Assume now that all graphs are triangle-divisible.
DefinitionAn absorber is a graph A such that A∪R has a triangle decompositionfor any sparse graph R.An absorber for a graph R is a graph AR such that AR and AR ∪R bothhave a triangle decomposition.
Approach: take A to be union of AR over all possible sparseremainders R.Far more than n2 possibilities for R, so no hope of finding oneabsorber for each R.
![Page 28: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/28.jpg)
Planning ahead
Plan1 Take out highly structured subgraph A.2 Take out many triangles to leave a sparse remainder R.3 Use ‘structure’ of A to decompose A∪R into triangles.
Assume now that all graphs are triangle-divisible.
DefinitionAn absorber is a graph A such that A∪R has a triangle decompositionfor any sparse graph R.
An absorber for a graph R is a graph AR such that AR and AR ∪R bothhave a triangle decomposition.
Approach: take A to be union of AR over all possible sparseremainders R.Far more than n2 possibilities for R, so no hope of finding oneabsorber for each R.
![Page 29: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/29.jpg)
Planning ahead
Plan1 Take out highly structured subgraph A.2 Take out many triangles to leave a sparse remainder R.3 Use ‘structure’ of A to decompose A∪R into triangles.
Assume now that all graphs are triangle-divisible.
DefinitionAn absorber is a graph A such that A∪R has a triangle decompositionfor any sparse graph R.An absorber for a graph R is a graph AR such that AR and AR ∪R bothhave a triangle decomposition.
Approach: take A to be union of AR over all possible sparseremainders R.
Far more than n2 possibilities for R, so no hope of finding oneabsorber for each R.
![Page 30: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/30.jpg)
Planning ahead
Plan1 Take out highly structured subgraph A.2 Take out many triangles to leave a sparse remainder R.3 Use ‘structure’ of A to decompose A∪R into triangles.
Assume now that all graphs are triangle-divisible.
DefinitionAn absorber is a graph A such that A∪R has a triangle decompositionfor any sparse graph R.An absorber for a graph R is a graph AR such that AR and AR ∪R bothhave a triangle decomposition.
Approach: take A to be union of AR over all possible sparseremainders R.Far more than n2 possibilities for R, so no hope of finding oneabsorber for each R.
![Page 31: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/31.jpg)
Absorbers
AimReduce the number of possible remainders R.
V1 V2 V nm
Let m be a large integer and equipartition the vertex set intoV1, . . . ,V n
meach of size m. Can we ensure that every edge of R is
contained within some Vi?
LemmaYes.
Let Ri be the part of R contained within Vi .For each i , there are at most 2(
m2) possibilities for Ri .
So we only need to find 2(m2)n/m = O(n) absorbers.
![Page 32: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/32.jpg)
Absorbers
AimReduce the number of possible remainders R.
V1 V2 V nm
Let m be a large integer and equipartition the vertex set intoV1, . . . ,V n
meach of size m. Can we ensure that every edge of R is
contained within some Vi?
LemmaYes.
Let Ri be the part of R contained within Vi .For each i , there are at most 2(
m2) possibilities for Ri .
So we only need to find 2(m2)n/m = O(n) absorbers.
![Page 33: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/33.jpg)
Absorbers
AimReduce the number of possible remainders R.
V1 V2 V nm
Let m be a large integer and equipartition the vertex set intoV1, . . . ,V n
meach of size m. Can we ensure that every edge of R is
contained within some Vi?
LemmaYes.
Let Ri be the part of R contained within Vi .For each i , there are at most 2(
m2) possibilities for Ri .
So we only need to find 2(m2)n/m = O(n) absorbers.
![Page 34: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/34.jpg)
Transformers
Definition
A graph T is an (H1,H2)-transformer if both H1∪T and T ∪H2 havetriangle decompositions.
Proposition
If H2 has a triangle decomposition, then T ∪H2 is an absorber for H1.
Write H1↔ H2 if there is an (H1,H2)-transformer.
Useful fact↔ is symmetric
↔ is transitive—if H1↔ H2↔ H3, then H1↔ H3
![Page 35: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/35.jpg)
Transformers
Definition
A graph T is an (H1,H2)-transformer if both H1∪T and T ∪H2 havetriangle decompositions.
Proposition
If H2 has a triangle decomposition, then T ∪H2 is an absorber for H1.
Write H1↔ H2 if there is an (H1,H2)-transformer.
Useful fact↔ is symmetric
↔ is transitive—if H1↔ H2↔ H3, then H1↔ H3
![Page 36: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/36.jpg)
Transformers
Definition
A graph T is an (H1,H2)-transformer if both H1∪T and T ∪H2 havetriangle decompositions.
Proposition
If H2 has a triangle decomposition, then T ∪H2 is an absorber for H1.
Write H1↔ H2 if there is an (H1,H2)-transformer.
Useful fact↔ is symmetric
↔ is transitive—if H1↔ H2↔ H3, then H1↔ H3
![Page 37: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/37.jpg)
Transformers
Definition
A graph T is an (H1,H2)-transformer if both H1∪T and T ∪H2 havetriangle decompositions.
Proposition
If H2 has a triangle decomposition, then T ∪H2 is an absorber for H1.
Write H1↔ H2 if there is an (H1,H2)-transformer.
Useful fact↔ is symmetric
↔ is transitive—if H1↔ H2↔ H3, then H1↔ H3
![Page 38: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/38.jpg)
Moving
Definition
A graph T is an (H1,H2)-transformer if both H1∪T and T ∪H2 havetriangle decompositions.
Suppose H1 is isomorphic to H2.
H2 = C9
H1 = C9
This allows us to ‘move graphs around’.
![Page 39: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/39.jpg)
Moving
Definition
A graph T is an (H1,H2)-transformer if both H1∪T and T ∪H2 havetriangle decompositions.
Suppose H1 is isomorphic to H2.
H2 = C9
H1 = C9
TTTTTTTTT
This allows us to ‘move graphs around’.
![Page 40: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/40.jpg)
Moving
Definition
A graph T is an (H1,H2)-transformer if both H1∪T and T ∪H2 havetriangle decompositions.
Suppose H1 is isomorphic to H2.
H2 = C9
H1 = C9
TTTTTTTTT
This allows us to ‘move graphs around’.
![Page 41: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/41.jpg)
Moving
Definition
A graph T is an (H1,H2)-transformer if both H1∪T and T ∪H2 havetriangle decompositions.
Suppose H1 is isomorphic to H2.
H2 = C9
H1 = C9
TTTTTTTTT
This allows us to ‘move graphs around’.
![Page 42: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/42.jpg)
Moving
Definition
A graph T is an (H1,H2)-transformer if both H1∪T and T ∪H2 havetriangle decompositions.
Suppose H1 is isomorphic to H2.
H2 = C9
H1 = C9
TTTTTTTTT
This allows us to ‘move graphs around’.
![Page 43: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/43.jpg)
Identifying vertices
Identify the green vertices of H2.
H2
H1
T
Note that T is still an (H1,H2)-transformer.So we can ‘identify vertices’ (providing no multiple edges are created).Since↔ is symmetric, we can ‘split vertices’ (providing the resultinggraph is still triangle-divisible).
![Page 44: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/44.jpg)
Identifying vertices
Identify the green vertices of H2.
H2
H1
T
Note that T is still an (H1,H2)-transformer.So we can ‘identify vertices’ (providing no multiple edges are created).
Since↔ is symmetric, we can ‘split vertices’ (providing the resultinggraph is still triangle-divisible).
![Page 45: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/45.jpg)
Identifying vertices
Identify the green vertices of H2.
H2
H1
T
Note that T is still an (H1,H2)-transformer.So we can ‘identify vertices’ (providing no multiple edges are created).Since↔ is symmetric, we can ‘split vertices’ (providing the resultinggraph is still triangle-divisible).
![Page 46: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/46.jpg)
Subdividing an edge
Let xy be an edge.
1 Attach a triangle to x2 Split the vertex x .
x y
z w
We can subdivide xy into a path of length 4.
So we can
identify vertices;
split a vertex;
subdivide an edge.
![Page 47: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/47.jpg)
Subdividing an edge
Let xy be an edge.1 Attach a triangle to x
2 Split the vertex x .
x y
z wz w
We can subdivide xy into a path of length 4.
So we can
identify vertices;
split a vertex;
subdivide an edge.
![Page 48: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/48.jpg)
Subdividing an edge
Let xy be an edge.1 Attach a triangle to x2 Split the vertex x .
x y
z wz w
We can subdivide xy into a path of length 4.
So we can
identify vertices;
split a vertex;
subdivide an edge.
![Page 49: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/49.jpg)
Subdividing an edge
Let xy be an edge.1 Attach a triangle to x2 Split the vertex x .
x y
z wz w
↔x ′′x ′ y
z w
We can subdivide xy into a path of length 4.
So we can
identify vertices;
split a vertex;
subdivide an edge.
![Page 50: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/50.jpg)
Subdividing an edge
Let xy be an edge.1 Attach a triangle to x2 Split the vertex x .
x y
z wz w
↔x ′′x ′ y
z w
We can subdivide xy into a path of length 4.
So we can
identify vertices;
split a vertex;
subdivide an edge.
![Page 51: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/51.jpg)
Constructing an absorber for a given H
Suppose that e(H) = 3k .
1 Subdivide all edges of H.2 Identify all original vertices of H.3 Let J be a union of k vertex-disjoint triangles.4 Subdivide all edges of J.5 Identify all original vertices of J.6 Since↔ is transitive, H↔ J.
Thus H has an absorber.
H
![Page 52: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/52.jpg)
Constructing an absorber for a given H
Suppose that e(H) = 3k .1 Subdivide all edges of H.
2 Identify all original vertices of H.3 Let J be a union of k vertex-disjoint triangles.4 Subdivide all edges of J.5 Identify all original vertices of J.6 Since↔ is transitive, H↔ J.
Thus H has an absorber.
H
![Page 53: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/53.jpg)
Constructing an absorber for a given H
Suppose that e(H) = 3k .1 Subdivide all edges of H.2 Identify all original vertices of H.
3 Let J be a union of k vertex-disjoint triangles.4 Subdivide all edges of J.5 Identify all original vertices of J.6 Since↔ is transitive, H↔ J.
Thus H has an absorber.
H
↔
L
3k
![Page 54: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/54.jpg)
Constructing an absorber for a given H
Suppose that e(H) = 3k .1 Subdivide all edges of H.2 Identify all original vertices of H.3 Let J be a union of k vertex-disjoint triangles.
4 Subdivide all edges of J.5 Identify all original vertices of J.6 Since↔ is transitive, H↔ J.
Thus H has an absorber.
H
↔
L
3k J
![Page 55: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/55.jpg)
Constructing an absorber for a given H
Suppose that e(H) = 3k .1 Subdivide all edges of H.2 Identify all original vertices of H.3 Let J be a union of k vertex-disjoint triangles.4 Subdivide all edges of J.
5 Identify all original vertices of J.6 Since↔ is transitive, H↔ J.
Thus H has an absorber.
H
↔
L
3k J
![Page 56: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/56.jpg)
Constructing an absorber for a given H
Suppose that e(H) = 3k .1 Subdivide all edges of H.2 Identify all original vertices of H.3 Let J be a union of k vertex-disjoint triangles.4 Subdivide all edges of J.5 Identify all original vertices of J.
6 Since↔ is transitive, H↔ J.Thus H has an absorber.
H
↔
L
3k ↔ J
![Page 57: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/57.jpg)
Constructing an absorber for a given H
Suppose that e(H) = 3k .1 Subdivide all edges of H.2 Identify all original vertices of H.3 Let J be a union of k vertex-disjoint triangles.4 Subdivide all edges of J.5 Identify all original vertices of J.6 Since↔ is transitive, H↔ J.
Thus H has an absorber.
H
↔
L
3k ↔ J
![Page 58: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/58.jpg)
Open problem: better fractional thresholds
Theorem (Barber, Kuhn, Lo, Osthus 2014+)
Every large triangle-divisible graph G with δ (G)≥ (δfrac(K3)+o(1))nhas a triangle decomposition.
Problem
Determine δfrac(F), i.e. the minimum degree threshold for a graph G tohave a fractional F -decomposition.
• For triangles, showing that δfrac(K3) = 3/4 could be combined withour results to show the actual ‘decomposition threshold’ is(3/4+o(1))n.• Actually, showing that 3n/4 guarantees ‘(fractional) almostdecomposition’ would suffice.
![Page 59: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/59.jpg)
Open problem: better fractional thresholds
Theorem (Barber, Kuhn, Lo, Osthus 2014+)
Every large triangle-divisible graph G with δ (G)≥ (δfrac(K3)+o(1))nhas a triangle decomposition.
Problem
Determine δfrac(F), i.e. the minimum degree threshold for a graph G tohave a fractional F -decomposition.
• For triangles, showing that δfrac(K3) = 3/4 could be combined withour results to show the actual ‘decomposition threshold’ is(3/4+o(1))n.• Actually, showing that 3n/4 guarantees ‘(fractional) almostdecomposition’ would suffice.
![Page 60: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/60.jpg)
Decompositions into even cycles
Theorem (Barber, Kuhn, Lo, Osthus 2014+)
For n sufficiently large, every C`-divisible graph G on n vertices with
δ (G)≥
{(23 +o(1)
)n if `= 4,(
12 +o(1)
)n if `≥ 6 is even,
has a C`-decomposition.
asymptotically best possible
`≥ 6 even`≥ 6 even
neither component is C`-divisible, but entire graph is
δ = n/2−1
n/2n/2
![Page 61: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/61.jpg)
Decompositions into even cycles
Theorem (Barber, Kuhn, Lo, Osthus 2014+)
For n sufficiently large, every C`-divisible graph G on n vertices with
δ (G)≥
{(23 +o(1)
)n if `= 4,(
12 +o(1)
)n if `≥ 6 is even,
has a C`-decomposition.
asymptotically best possible
`= 4 (Kahn & Winkler)
δ = 3n/5−1, odd number of edges in blown-up C5
n/5
n/5
n/5 n/5
n/5
![Page 62: web.mat.bham.ac.uk › D.Osthus › Decomp_banff.pdf · Decompositions of dense graphs into small subgraphsTriangle decompositions of graphs of large minimum degree Conjecture (Nash-Williams](https://reader033.fdocuments.us/reader033/viewer/2022041819/5e5cb105cf704511cd5072c2/html5/thumbnails/62.jpg)
Decompositions into even cycles
Theorem (Barber, Kuhn, Lo, Osthus 2014+)
For n sufficiently large, every C`-divisible graph G on n vertices with
δ (G)≥
{(23 +o(1)
)n if `= 4,(
12 +o(1)
)n if `≥ 6 is even,
has a C`-decomposition.
asymptotically best possible
`= 4 (Taylor)
n/3n/3
A
n/3
δ = 2n/3−2
every C4 has even number of edges inside A, but e(A) is odd