A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager...
Transcript of A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager...
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A Ramsey Version of Graph Saturation
Mike Ferrara Jaehoon Kim Elyse Yeager
Midwest Conference on Combinatorics and Combinatorical Computing,University of Nevada, Las Vegas
24 October 2014
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9
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Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not asubgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Definitions
The saturation number sat(n;H) of a forbidden graph H is the smallestnumber of edges over all n-vertex graphs that are H-saturated.
Definitions
Given a forbidden family of graphs F , a graph G is F-saturated if nomember of F is a subgraph of G , but for every e ∈ G , some member of Fis a subgraph of G + e.The saturation number sat(n;F) is the smallest number of edges overall n-vertex graphs that are F-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 2 / 9
![Page 3: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/3.jpg)
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not asubgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Definitions
The saturation number sat(n;H) of a forbidden graph H is the smallestnumber of edges over all n-vertex graphs that are H-saturated.
Definitions
Given a forbidden family of graphs F , a graph G is F-saturated if nomember of F is a subgraph of G , but for every e ∈ G , some member of Fis a subgraph of G + e.The saturation number sat(n;F) is the smallest number of edges overall n-vertex graphs that are F-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 2 / 9
![Page 4: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/4.jpg)
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not asubgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Definitions
The saturation number sat(n;H) of a forbidden graph H is the smallestnumber of edges over all n-vertex graphs that are H-saturated.
Definitions
Given a forbidden family of graphs F , a graph G is F-saturated if nomember of F is a subgraph of G , but for every e ∈ G , some member of Fis a subgraph of G + e.The saturation number sat(n;F) is the smallest number of edges overall n-vertex graphs that are F-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 2 / 9
![Page 5: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/5.jpg)
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not asubgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Definitions
The saturation number sat(n;H) of a forbidden graph H is the smallestnumber of edges over all n-vertex graphs that are H-saturated.
Definitions
Given a forbidden family of graphs F , a graph G is F-saturated if nomember of F is a subgraph of G , but for every e ∈ G , some member of Fis a subgraph of G + e.The saturation number sat(n;F) is the smallest number of edges overall n-vertex graphs that are F-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 2 / 9
![Page 6: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/6.jpg)
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not asubgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Definitions
The saturation number sat(n;H) of a forbidden graph H is the smallestnumber of edges over all n-vertex graphs that are H-saturated.
Definitions
Given a forbidden family of graphs F , a graph G is F-saturated if nomember of F is a subgraph of G , but for every e ∈ G , some member of Fis a subgraph of G + e.The saturation number sat(n;F) is the smallest number of edges overall n-vertex graphs that are F-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 2 / 9
![Page 7: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/7.jpg)
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not asubgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Definitions
The saturation number sat(n;H) of a forbidden graph H is the smallestnumber of edges over all n-vertex graphs that are H-saturated.
Definitions
Given a forbidden family of graphs F , a graph G is F-saturated if nomember of F is a subgraph of G , but for every e ∈ G , some member of Fis a subgraph of G + e.The saturation number sat(n;F) is the smallest number of edges overall n-vertex graphs that are F-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 2 / 9
![Page 8: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/8.jpg)
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not asubgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Definitions
The saturation number sat(n;H) of a forbidden graph H is the smallestnumber of edges over all n-vertex graphs that are H-saturated.
Definitions
Given a forbidden family of graphs F , a graph G is F-saturated if nomember of F is a subgraph of G , but for every e ∈ G , some member of Fis a subgraph of G + e.The saturation number sat(n;F) is the smallest number of edges overall n-vertex graphs that are F-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 2 / 9
![Page 9: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/9.jpg)
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not asubgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Definitions
The saturation number sat(n;H) of a forbidden graph H is the smallestnumber of edges over all n-vertex graphs that are H-saturated.
Definitions
Given a forbidden family of graphs F , a graph G is F-saturated if nomember of F is a subgraph of G , but for every e ∈ G , some member of Fis a subgraph of G + e.The saturation number sat(n;F) is the smallest number of edges overall n-vertex graphs that are F-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 2 / 9
![Page 10: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/10.jpg)
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not asubgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Definitions
The saturation number sat(n;H) of a forbidden graph H is the smallestnumber of edges over all n-vertex graphs that are H-saturated.
Definitions
Given a forbidden family of graphs F , a graph G is F-saturated if nomember of F is a subgraph of G , but for every e ∈ G , some member of Fis a subgraph of G + e.The saturation number sat(n;F) is the smallest number of edges overall n-vertex graphs that are F-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 2 / 9
![Page 11: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/11.jpg)
Ramsey-Minimal Families
Definitions
A graph G is (H1, . . . ,Hk)-Ramsey minimal if G → (H1, . . . ,Hk) but forany e ∈ E (G ), G − e 6→ (H1, . . . ,Hk).
Example: K6 → (K3,K3), but K6 − e 6→ (K3,K3).
Definitions
Rmin(H1, . . . ,Hk) = Rmin = {G : G is (H1, . . . ,Hk)-Ramsey minimal}
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 3 / 9
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Ramsey-Minimal Families
Definitions
A graph G is (H1, . . . ,Hk)-Ramsey minimal if G → (H1, . . . ,Hk) but forany e ∈ E (G ), G − e 6→ (H1, . . . ,Hk).
Example: K6 → (K3,K3), but K6 − e 6→ (K3,K3).
Definitions
Rmin(H1, . . . ,Hk) = Rmin = {G : G is (H1, . . . ,Hk)-Ramsey minimal}
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 3 / 9
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Ramsey-Minimal Families
Definitions
A graph G is (H1, . . . ,Hk)-Ramsey minimal if G → (H1, . . . ,Hk) but forany e ∈ E (G ), G − e 6→ (H1, . . . ,Hk).
Example: K6 → (K3,K3), but K6 − e 6→ (K3,K3).
Definitions
Rmin(H1, . . . ,Hk) = Rmin = {G : G is (H1, . . . ,Hk)-Ramsey minimal}
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 3 / 9
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Ramsey-Minimal Families
Definitions
A graph G is (H1, . . . ,Hk)-Ramsey minimal if G → (H1, . . . ,Hk) but forany e ∈ E (G ), G − e 6→ (H1, . . . ,Hk).
Example: K6 → (K3,K3), but K6 − e 6→ (K3,K3).
Definitions
Rmin(H1, . . . ,Hk) = Rmin = {G : G is (H1, . . . ,Hk)-Ramsey minimal}
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 3 / 9
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Ramsey-Minimal Families
Definitions
A graph G is (H1, . . . ,Hk)-Ramsey minimal if G → (H1, . . . ,Hk) but forany e ∈ E (G ), G − e 6→ (H1, . . . ,Hk).
Example: K6 → (K3,K3), but K6 − e 6→ (K3,K3).
Definitions
Rmin(H1, . . . ,Hk) = Rmin = {G : G is (H1, . . . ,Hk)-Ramsey minimal}
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 3 / 9
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Ramsey-Minimal Families
Definitions
A graph G is (H1, . . . ,Hk)-Ramsey minimal if G → (H1, . . . ,Hk) but forany e ∈ E (G ), G − e 6→ (H1, . . . ,Hk).
Example: K6 → (K3,K3), but K6 − e 6→ (K3,K3).
Definitions
Rmin(H1, . . . ,Hk) = Rmin = {G : G is (H1, . . . ,Hk)-Ramsey minimal}
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 3 / 9
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Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
A graph G is Rmin(H1, . . . ,Hk) saturated if and only if:
Example: the graph below is Rmin(K3,K3)-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 4 / 9
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Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
A graph G is Rmin(H1, . . . ,Hk) saturated if and only if:
Example: the graph below is Rmin(K3,K3)-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 4 / 9
![Page 19: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/19.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
A graph G is Rmin(H1, . . . ,Hk) saturated if and only if:
G 6→ (H1, . . . ,Hk)
For any e ∈ G , G + e → (H1, . . . ,Hk).
Example: the graph below is Rmin(K3,K3)-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 4 / 9
![Page 20: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/20.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
A graph G is Rmin(H1, . . . ,Hk) saturated if and only if:
G 6→ (H1, . . . ,Hk)
For any e ∈ G , G + e → (H1, . . . ,Hk).
Example: the graph below is Rmin(K3,K3)-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 4 / 9
![Page 21: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/21.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
A graph G is Rmin(H1, . . . ,Hk) saturated if and only if:
G 6→ (H1, . . . ,Hk)
For any e ∈ G , G + e → (H1, . . . ,Hk).
Example: the graph below is Rmin(K3,K3)-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 4 / 9
![Page 22: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/22.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
A graph G is Rmin(H1, . . . ,Hk) saturated if and only if:
G 6→ (H1, . . . ,Hk)
For any e ∈ G , G + e → (H1, . . . ,Hk).
Example: the graph below is Rmin(K3,K3)-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 4 / 9
![Page 23: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/23.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
A graph G is Rmin(H1, . . . ,Hk) saturated if and only if:
G 6→ (H1, . . . ,Hk)
For any e ∈ G , G + e → (H1, . . . ,Hk).
Example: the graph below is Rmin(K3,K3)-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 4 / 9
![Page 24: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/24.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
A graph G is Rmin(H1, . . . ,Hk) saturated if and only if:
G 6→ (H1, . . . ,Hk)
For any e ∈ G , G + e → (H1, . . . ,Hk).
Example: the graph below is Rmin(K3,K3)-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 4 / 9
![Page 25: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/25.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
A graph G is Rmin(H1, . . . ,Hk) saturated if and only if:
G 6→ (H1, . . . ,Hk)
For any e ∈ G , G + e → (H1, . . . ,Hk).
Example: the graph below is Rmin(K3,K3)-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 4 / 9
![Page 26: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/26.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
A graph G is Rmin(H1, . . . ,Hk) saturated if and only if:
G 6→ (H1, . . . ,Hk)
For any e ∈ G , G + e → (H1, . . . ,Hk).
Example: the graph below is Rmin(K3,K3)-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 4 / 9
![Page 27: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/27.jpg)
Saturation of Rmin(Kk1, . . . ,Kkt)
Example
Let r := r(k1, . . . , kt) be the Ramsey number of (Kk1 , . . . ,Kkt ). Then
Kr−2 ∨ Ks
is Rmin(Kk1 , . . . ,Kkt ) saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 5 / 9
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Saturation of Rmin(Kk1, . . . ,Kkt)
Example
Let r := r(k1, . . . , kt) be the Ramsey number of (Kk1 , . . . ,Kkt ). Then
Kr−2 ∨ Ks
is Rmin(Kk1 , . . . ,Kkt ) saturated.
Kr−2 n − (r − 2)
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 5 / 9
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Saturation of Rmin(Kk1, . . . ,Kkt)
Example
Let r := r(k1, . . . , kt) be the Ramsey number of (Kk1 , . . . ,Kkt ). Then
Kr−2 ∨ Ks
is Rmin(Kk1 , . . . ,Kkt ) saturated.
Kr−2 n − (r − 2)
Kr−2 ∨ Ks 6→ (Kk1 , . . . ,Kkt )Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 5 / 9
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Saturation of Rmin(Kk1, . . . ,Kkt)
Example
Let r := r(k1, . . . , kt) be the Ramsey number of (Kk1 , . . . ,Kkt ). Then
Kr−2 ∨ Ks
is Rmin(Kk1 , . . . ,Kkt ) saturated.
Kr−2 n − (r − 2)
Kr−2 ∨ Ks 6→ (Kk1 , . . . ,Kkt )Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 5 / 9
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Saturation of Rmin(Kk1, . . . ,Kkt)
Example
Let r := r(k1, . . . , kt) be the Ramsey number of (Kk1 , . . . ,Kkt ). Then
Kr−2 ∨ Ks
is Rmin(Kk1 , . . . ,Kkt ) saturated.
Kr−2 n − (r − 2)
Kr−2 ∨ Ks 6→ (Kk1 , . . . ,Kkt )Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 5 / 9
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Saturation of Rmin(Kk1, . . . ,Kkt)
Example
Let r := r(k1, . . . , kt) be the Ramsey number of (Kk1 , . . . ,Kkt ). Then
Kr−2 ∨ Ks
is Rmin(Kk1 , . . . ,Kkt ) saturated.
Kr−2 n − (r − 2)
Kr−2 ∨ Ks 6→ (Kk1 , . . . ,Kkt )Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 5 / 9
![Page 33: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/33.jpg)
Saturation of Rmin(Kk1, . . . ,Kkt)
Example
Let r := r(k1, . . . , kt) be the Ramsey number of (Kk1 , . . . ,Kkt ). Then
Kr−2 ∨ Ks
is Rmin(Kk1 , . . . ,Kkt ) saturated.
Kr−2 n − (r − 2)
Kr−2 ∨ Ks 6→ (Kk1 , . . . ,Kkt )Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 5 / 9
![Page 34: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/34.jpg)
Saturation of Rmin(Kk1, . . . ,Kkt)
Example
Let r := r(k1, . . . , kt) be the Ramsey number of (Kk1 , . . . ,Kkt ). Then
Kr−2 ∨ Ks
is Rmin(Kk1 , . . . ,Kkt ) saturated.
Kr−2 n − (r − 2)
Kr−2 ∨ Ks 6→ (Kk1 , . . . ,Kkt ) Kr−2 ∨Ks + e → (Kk1 , . . . ,Kkt )Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 5 / 9
![Page 35: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/35.jpg)
Saturation of Rmin(Kk1, . . . ,Kkt)
Example
Let r := r(k1, . . . , kt) be the Ramsey number of (Kk1 , . . . ,Kkt ). Then
Kr−2 ∨ Ks
is Rmin(Kk1 , . . . ,Kkt ) saturated.
Kr−2 n − (r − 2)
Kr−2 ∨ Ks 6→ (Kk1 , . . . ,Kkt ) Kr−2 ∨Ks + e → (Kk1 , . . . ,Kkt )Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 5 / 9
![Page 36: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/36.jpg)
Saturation of Rmin(Kk1, . . . ,Kkt)
Example
Let r := r(k1, . . . , kt) be the Ramsey number of (Kk1 , . . . ,Kkt ). Then
Kr−2 ∨ Ks
is Rmin(Kk1 , . . . ,Kkt ) saturated.
Kr−2 n − (r − 2)
Kr−2 ∨ Ks 6→ (Kk1 , . . . ,Kkt ) Kr−2 ∨Ks + e → (Kk1 , . . . ,Kkt )Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 5 / 9
![Page 37: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/37.jpg)
Saturation of Rmin(Kk1, . . . ,Kkt)
Example
Let r := r(k1, . . . , kt) be the Ramsey number of (Kk1 , . . . ,Kkt ). Then
Kr−2 ∨ Ks
is Rmin(Kk1 , . . . ,Kkt ) saturated.
Corollary
sat(n;Rmin(Kk1 , . . . ,Kkt )) ≤(r−2
2
)+ (r − 2)(n − r + 2) when n ≥ r
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 5 / 9
![Page 38: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/38.jpg)
Saturation of Rmin(Kk1, . . . ,Kkt)
Example
Let r := r(k1, . . . , kt) be the Ramsey number of (Kk1 , . . . ,Kkt ). Then
Kr−2 ∨ Ks
is Rmin(Kk1 , . . . ,Kkt ) saturated.
Corollary
sat(n;Rmin(Kk1 , . . . ,Kkt )) ≤(r−2
2
)+ (r − 2)(n − r + 2) when n ≥ r
Hanson-Toft Conjecture, 1987
sat(n;Rmin(Kk1 , . . . ,Kkt )) =
{ (n2
)n < r(r−2
2
)+ (r − 2)(n − r + 2) n ≥ r
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 5 / 9
![Page 39: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/39.jpg)
Hanson-Toft
Hanson-Toft Conjecture
sat(n;Rmin(Kk1 , . . . ,Kkt )) =
{ (n2
)n < r(r−2
2
)+ (r − 2)(n − r + 2) n ≥ r
Chen, Ferrara, Gould, Magnant, Schmitt; 2011
sat(n;Rmin(K3,K3)) =
{ (n2
)n < 6 = r
4n − 10 n ≥ 56
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 6 / 9
![Page 40: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/40.jpg)
Hanson-Toft
Hanson-Toft Conjecture
sat(n;Rmin(Kk1 , . . . ,Kkt )) =
{ (n2
)n < r(r−2
2
)+ (r − 2)(n − r + 2) n ≥ r
Chen, Ferrara, Gould, Magnant, Schmitt; 2011
sat(n;Rmin(K3,K3)) =
{ (n2
)n < 6 = r
4n − 10 n ≥ 56
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 6 / 9
![Page 41: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/41.jpg)
Hanson-Toft
Hanson-Toft Conjecture
sat(n;Rmin(Kk1 , . . . ,Kkt )) =
{ (n2
)n < r(r−2
2
)+ (r − 2)(n − r + 2) n ≥ r
Chen, Ferrara, Gould, Magnant, Schmitt; 2011
sat(n;Rmin(K3,K3)) =
{ (n2
)n < 6 = r
4n − 10 n ≥ 56
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 6 / 9
![Page 42: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/42.jpg)
Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
(5K2, 5K2, 5K2, 5K2)
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 7 / 9
![Page 43: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/43.jpg)
Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
(5K2, 5K2, 5K2, 5K2)
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 7 / 9
![Page 44: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/44.jpg)
Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
(5K2, 5K2, 5K2, 5K2)
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 7 / 9
![Page 45: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/45.jpg)
Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
(5K2, 5K2, 5K2, 5K2)
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 7 / 9
![Page 46: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/46.jpg)
Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
(5K2, 5K2, 5K2, 5K2)
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 7 / 9
![Page 47: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/47.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 48: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/48.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 49: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/49.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 50: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/50.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 51: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/51.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 52: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/52.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 53: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/53.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 54: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/54.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 55: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/55.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 56: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/56.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 57: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/57.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 58: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/58.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 59: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/59.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 60: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/60.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 61: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/61.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 62: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/62.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 63: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/63.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 64: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/64.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 65: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/65.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 66: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/66.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 67: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/67.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 68: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/68.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 69: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/69.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 70: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/70.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 71: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/71.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 72: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/72.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 73: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/73.jpg)
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Example: Forbidden graphs (3K2, 3K2).
good coloring
make red-heavy
take red subgraph
This (uncolored) subgraph is 3K2-saturated.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 8 / 9
![Page 74: A Ramsey Version of Graph Saturationelyse/Talk_UNLV14_2.pdf · 24 October 2014 Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 1 / 9. Graph Saturation](https://reader033.fdocuments.in/reader033/viewer/2022060522/60513084f4455c1c6444b2f6/html5/thumbnails/74.jpg)
Thanks for Listening!
- G. Chen, M. Ferrara, R. Gould, C. Magnant, J. Schmitt,Saturation numbers for families of Ramsey-minimal graphs, J.Combin. 2 (2011) 435-455.
- M. Ferrara, J. Kim, E. Yeager, Ramsey-minimal saturationnumbers for matchings, Discrete Math. 322 (2014) 26-30.
- A. Galluccio, M. Simonovits, G. Simonyi, On the structure ofco-critical graphs, In: Graph Theory, Combinatorics and Algorithms,Vol. 1, 2 (Kalamazoo, MI, 1992). Wiley-Intersci. Publ., Wiley, NewYork, 1053-1071.
- D. Hanson, B. Toft, Edge-colored saturated graphs, J. GraphTheory 11 (1987), no. 2, 191-196.
- T. Szabo, On nearly regular co-critical graphs, Discrete Math. 160(1996) 279-281.
Ferrara-Kim-Yeager (UCD, UIUC) Ramsey Version of Saturation 24 October 2014 9 / 9