saturation number of ramsey-minimal familieselyse/yeager_ramsey_saturation_detroit.… ·...
TRANSCRIPT
![Page 1: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/1.jpg)
Saturation Number of Ramsey-Minimal Families
Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2
1University of Colorado-Denver
2University of Illinois at Urbana-Champaign
MIGHTYUniversity of Detroit Mercy
29 March 2014
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 1 / 11
![Page 2: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/2.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) Saturation of Ramsey-Minimal Families 29 March 2014 2 / 11
![Page 3: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/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) Saturation of Ramsey-Minimal Families 29 March 2014 2 / 11
![Page 4: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/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) Saturation of Ramsey-Minimal Families 29 March 2014 2 / 11
![Page 5: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/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) Saturation of Ramsey-Minimal Families 29 March 2014 2 / 11
![Page 6: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/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) Saturation of Ramsey-Minimal Families 29 March 2014 2 / 11
![Page 7: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/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) Saturation of Ramsey-Minimal Families 29 March 2014 2 / 11
![Page 8: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/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) Saturation of Ramsey-Minimal Families 29 March 2014 2 / 11
![Page 9: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/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) Saturation of Ramsey-Minimal Families 29 March 2014 2 / 11
![Page 10: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/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) Saturation of Ramsey-Minimal Families 29 March 2014 2 / 11
![Page 11: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/11.jpg)
Ramsey-Minimal Families
Definitions
Given ”forbidden” graphs H1, . . . ,Hk , and any graph G , we writeG→ (H1, . . . ,Hk) if any k coloring of E (G ) contains a monochromaticcopy of Hi in color i , for some i .
Famous Example: K6 → (K3,K3), but K5 6→ (K3,K3)
Definitions
A graph G is (H1, . . . ,Hk)-Ramsey minimal if G → (H1, . . . ,Hk) but forany e ∈ E (G ), G − e 6→ (H1, . . . ,Hk).
Less Famous Example: K6 is (K3,K3)-Ramsey Minimal.
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 3 / 11
![Page 12: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/12.jpg)
Ramsey-Minimal Families
Definitions
Given ”forbidden” graphs H1, . . . ,Hk , and any graph G , we writeG→ (H1, . . . ,Hk) if any k coloring of E (G ) contains a monochromaticcopy of Hi in color i , for some i .
Famous Example: K6 → (K3,K3), but K5 6→ (K3,K3)
Definitions
A graph G is (H1, . . . ,Hk)-Ramsey minimal if G → (H1, . . . ,Hk) but forany e ∈ E (G ), G − e 6→ (H1, . . . ,Hk).
Less Famous Example: K6 is (K3,K3)-Ramsey Minimal.
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 3 / 11
![Page 13: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/13.jpg)
Ramsey-Minimal Families
Definitions
Given ”forbidden” graphs H1, . . . ,Hk , and any graph G , we writeG→ (H1, . . . ,Hk) if any k coloring of E (G ) contains a monochromaticcopy of Hi in color i , for some i .
Famous Example: K6 → (K3,K3), but K5 6→ (K3,K3)
Definitions
A graph G is (H1, . . . ,Hk)-Ramsey minimal if G → (H1, . . . ,Hk) but forany e ∈ E (G ), G − e 6→ (H1, . . . ,Hk).
Less Famous Example: K6 is (K3,K3)-Ramsey Minimal.
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 3 / 11
![Page 14: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/14.jpg)
Ramsey-Minimal Families
Definitions
Given ”forbidden” graphs H1, . . . ,Hk , and any graph G , we writeG→ (H1, . . . ,Hk) if any k coloring of E (G ) contains a monochromaticcopy of Hi in color i , for some i .
Famous Example: K6 → (K3,K3), but K5 6→ (K3,K3)
Definitions
A graph G is (H1, . . . ,Hk)-Ramsey minimal if G → (H1, . . . ,Hk) but forany e ∈ E (G ), G − e 6→ (H1, . . . ,Hk).
Less Famous Example: K6 is (K3,K3)-Ramsey Minimal.
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 3 / 11
![Page 15: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/15.jpg)
Definitions
A graph G is (H1, . . . ,Hk)-Ramsey minimal if G → (H1, . . . ,Hk) but forany e ∈ E (G ), G − e 6→ (H1, . . . ,Hk).
Less Famous Example: K6 is (K3,K3)-Ramsey Minimal.
K6 ∈ Rmin(K3,K3)
Definitions
Rmin(H1, . . . ,Hk) = Rmin = {G : G is (H1, . . . ,Hk)-Ramsey minimal}
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 4 / 11
![Page 16: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/16.jpg)
Definitions
A graph G is (H1, . . . ,Hk)-Ramsey minimal if G → (H1, . . . ,Hk) but forany e ∈ E (G ), G − e 6→ (H1, . . . ,Hk).
Less Famous Example: K6 is (K3,K3)-Ramsey Minimal.
K6 ∈ Rmin(K3,K3)
Definitions
Rmin(H1, . . . ,Hk) = Rmin = {G : G is (H1, . . . ,Hk)-Ramsey minimal}
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 4 / 11
![Page 17: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/17.jpg)
Definitions
A graph G is (H1, . . . ,Hk)-Ramsey minimal if G → (H1, . . . ,Hk) but forany e ∈ E (G ), G − e 6→ (H1, . . . ,Hk).
Less Famous Example: K6 is (K3,K3)-Ramsey Minimal.
K6 ∈ Rmin(K3,K3)
Definitions
Rmin(H1, . . . ,Hk) = Rmin = {G : G is (H1, . . . ,Hk)-Ramsey minimal}
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 4 / 11
![Page 18: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/18.jpg)
Definitions
A graph G is (H1, . . . ,Hk)-Ramsey minimal if G → (H1, . . . ,Hk) but forany e ∈ E (G ), G − e 6→ (H1, . . . ,Hk).
Less Famous Example: K6 is (K3,K3)-Ramsey Minimal.
K6 ∈ Rmin(K3,K3)
Definitions
Rmin(H1, . . . ,Hk) = Rmin = {G : G is (H1, . . . ,Hk)-Ramsey minimal}
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 4 / 11
![Page 19: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/19.jpg)
Definitions
A graph G is (H1, . . . ,Hk)-Ramsey minimal if G → (H1, . . . ,Hk) but forany e ∈ E (G ), G − e 6→ (H1, . . . ,Hk).
Less Famous Example: K6 is (K3,K3)-Ramsey Minimal.
K6 ∈ Rmin(K3,K3)
Definitions
Rmin(H1, . . . ,Hk) = Rmin = {G : G is (H1, . . . ,Hk)-Ramsey minimal}
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 4 / 11
![Page 20: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/20.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
Suppose G is Rmin(H1, . . . ,Hk) saturated.
G has no subgraph that is (H1, . . . ,Hk)-Ramsey minimal
I G 6→ (H1, . . . ,Hk)Pf: If G → (H1, . . . ,Hk), we delete edges as long as the deletion doesnot cause an admissible coloring to exist
Adding any edge to G creates a subgraph that is(H1, . . . ,Hk)-Ramsey minimal
I For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Rmin(H1, . . . ,Hk) Saturation
G is Rmin(H1, . . . ,Hk) saturated iff
G 6→ (H1, . . . ,Hk)
For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 5 / 11
![Page 21: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/21.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
Suppose G is Rmin(H1, . . . ,Hk) saturated.
G has no subgraph that is (H1, . . . ,Hk)-Ramsey minimal
I G 6→ (H1, . . . ,Hk)Pf: If G → (H1, . . . ,Hk), we delete edges as long as the deletion doesnot cause an admissible coloring to exist
Adding any edge to G creates a subgraph that is(H1, . . . ,Hk)-Ramsey minimal
I For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Rmin(H1, . . . ,Hk) Saturation
G is Rmin(H1, . . . ,Hk) saturated iff
G 6→ (H1, . . . ,Hk)
For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 5 / 11
![Page 22: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/22.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
Suppose G is Rmin(H1, . . . ,Hk) saturated.
G has no subgraph that is (H1, . . . ,Hk)-Ramsey minimal
I G 6→ (H1, . . . ,Hk)Pf: If G → (H1, . . . ,Hk), we delete edges as long as the deletion doesnot cause an admissible coloring to exist
Adding any edge to G creates a subgraph that is(H1, . . . ,Hk)-Ramsey minimal
I For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Rmin(H1, . . . ,Hk) Saturation
G is Rmin(H1, . . . ,Hk) saturated iff
G 6→ (H1, . . . ,Hk)
For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 5 / 11
![Page 23: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/23.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
Suppose G is Rmin(H1, . . . ,Hk) saturated.
G has no subgraph that is (H1, . . . ,Hk)-Ramsey minimal
I G 6→ (H1, . . . ,Hk)Pf: If G → (H1, . . . ,Hk), we delete edges as long as the deletion doesnot cause an admissible coloring to exist
Adding any edge to G creates a subgraph that is(H1, . . . ,Hk)-Ramsey minimal
I For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Rmin(H1, . . . ,Hk) Saturation
G is Rmin(H1, . . . ,Hk) saturated iff
G 6→ (H1, . . . ,Hk)
For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 5 / 11
![Page 24: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/24.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
Suppose G is Rmin(H1, . . . ,Hk) saturated.
G has no subgraph that is (H1, . . . ,Hk)-Ramsey minimal
I G 6→ (H1, . . . ,Hk)Pf: If G → (H1, . . . ,Hk), we delete edges as long as the deletion doesnot cause an admissible coloring to exist
Adding any edge to G creates a subgraph that is(H1, . . . ,Hk)-Ramsey minimal
I For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Rmin(H1, . . . ,Hk) Saturation
G is Rmin(H1, . . . ,Hk) saturated iff
G 6→ (H1, . . . ,Hk)
For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 5 / 11
![Page 25: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/25.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
Suppose G is Rmin(H1, . . . ,Hk) saturated.
G has no subgraph that is (H1, . . . ,Hk)-Ramsey minimalI G 6→ (H1, . . . ,Hk)
Pf: If G → (H1, . . . ,Hk), we delete edges as long as the deletion doesnot cause an admissible coloring to exist
Adding any edge to G creates a subgraph that is(H1, . . . ,Hk)-Ramsey minimal
I For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Rmin(H1, . . . ,Hk) Saturation
G is Rmin(H1, . . . ,Hk) saturated iff
G 6→ (H1, . . . ,Hk)
For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 5 / 11
![Page 26: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/26.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
Suppose G is Rmin(H1, . . . ,Hk) saturated.
G has no subgraph that is (H1, . . . ,Hk)-Ramsey minimalI G 6→ (H1, . . . ,Hk)
Pf: If G → (H1, . . . ,Hk), we delete edges as long as the deletion doesnot cause an admissible coloring to exist
Adding any edge to G creates a subgraph that is(H1, . . . ,Hk)-Ramsey minimal
I For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Rmin(H1, . . . ,Hk) Saturation
G is Rmin(H1, . . . ,Hk) saturated iff
G 6→ (H1, . . . ,Hk)
For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 5 / 11
![Page 27: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/27.jpg)
Saturation of Ramsey-Minimal Families
Rmin(H1, . . . ,Hk) Saturation
Suppose G is Rmin(H1, . . . ,Hk) saturated.
G has no subgraph that is (H1, . . . ,Hk)-Ramsey minimalI G 6→ (H1, . . . ,Hk)
Pf: If G → (H1, . . . ,Hk), we delete edges as long as the deletion doesnot cause an admissible coloring to exist
Adding any edge to G creates a subgraph that is(H1, . . . ,Hk)-Ramsey minimal
I For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Rmin(H1, . . . ,Hk) Saturation
G is Rmin(H1, . . . ,Hk) saturated iff
G 6→ (H1, . . . ,Hk)
For any e ∈ E (G ), G + e → (H1, . . . ,Hk)
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 5 / 11
![Page 28: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/28.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) Saturation of Ramsey-Minimal Families 29 March 2014 6 / 11
![Page 29: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/29.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) Saturation of Ramsey-Minimal Families 29 March 2014 6 / 11
![Page 30: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/30.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 ∨ Ks 6→ (Kk1 , . . . ,Kkt )
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 6 / 11
![Page 31: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/31.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 ∨ Ks 6→ (Kk1 , . . . ,Kkt ) Kr−2 ∨Ks + e → (Kk1 , . . . ,Kkt )
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 6 / 11
![Page 32: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/32.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) Saturation of Ramsey-Minimal Families 29 March 2014 6 / 11
![Page 33: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/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.
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) Saturation of Ramsey-Minimal Families 29 March 2014 6 / 11
![Page 34: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/34.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) Saturation of Ramsey-Minimal Families 29 March 2014 7 / 11
![Page 35: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/35.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) Saturation of Ramsey-Minimal Families 29 March 2014 7 / 11
![Page 36: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/36.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) Saturation of Ramsey-Minimal Families 29 March 2014 7 / 11
![Page 37: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/37.jpg)
Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
(5K2, 5K2, 5K2, 5K2)
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 8 / 11
![Page 38: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/38.jpg)
Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
(5K2, 5K2, 5K2, 5K2)
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 8 / 11
![Page 39: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/39.jpg)
Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
(5K2, 5K2, 5K2, 5K2)
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 8 / 11
![Page 40: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/40.jpg)
Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
(5K2, 5K2, 5K2, 5K2)
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 8 / 11
![Page 41: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/41.jpg)
Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
(5K2, 5K2, 5K2, 5K2)
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 8 / 11
![Page 42: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/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) Saturation of Ramsey-Minimal Families 29 March 2014 8 / 11
![Page 43: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/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) Saturation of Ramsey-Minimal Families 29 March 2014 8 / 11
![Page 44: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/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) Saturation of Ramsey-Minimal Families 29 March 2014 8 / 11
![Page 45: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/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) Saturation of Ramsey-Minimal Families 29 March 2014 8 / 11
![Page 46: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/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) Saturation of Ramsey-Minimal Families 29 March 2014 8 / 11
![Page 47: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/47.jpg)
Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
(5K2, 5K2, 5K2, 5K2)
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 8 / 11
![Page 48: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/48.jpg)
Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
(5K2, 5K2, 5K2, 5K2)
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 8 / 11
![Page 49: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/49.jpg)
Sat Number of Ramsey-Minimal Families of Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
Corollary
sat(n;Rmin(k1K2 + · · ·+ ktK2)) ≤ 3(k1 + · · ·+ kt − t)when n ≥ 3(k1 + · · ·+ kt − t)
Ferrara, Kim, Y.; 2014
sat(n;Rmin(k1K2 + · · ·+ ktK2)) = 3(k1 + · · ·+ kt − t)when n > 3(k1 + · · ·+ kt − t)
Construction is generally unique: vertex-disjoint triangles with isolates.
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 9 / 11
![Page 50: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/50.jpg)
Sat Number of Ramsey-Minimal Families of Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
Corollary
sat(n;Rmin(k1K2 + · · ·+ ktK2)) ≤ 3(k1 + · · ·+ kt − t)when n ≥ 3(k1 + · · ·+ kt − t)
Ferrara, Kim, Y.; 2014
sat(n;Rmin(k1K2 + · · ·+ ktK2)) = 3(k1 + · · ·+ kt − t)when n > 3(k1 + · · ·+ kt − t)
Construction is generally unique: vertex-disjoint triangles with isolates.
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 9 / 11
![Page 51: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/51.jpg)
Sat Number of Ramsey-Minimal Families of Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
Corollary
sat(n;Rmin(k1K2 + · · ·+ ktK2)) ≤ 3(k1 + · · ·+ kt − t)when n ≥ 3(k1 + · · ·+ kt − t)
Ferrara, Kim, Y.; 2014
sat(n;Rmin(k1K2 + · · ·+ ktK2)) = 3(k1 + · · ·+ kt − t)when n > 3(k1 + · · ·+ kt − t)
Construction is generally unique: vertex-disjoint triangles with isolates.
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 9 / 11
![Page 52: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/52.jpg)
Sat Number of Ramsey-Minimal Families of Matchings
Example
(k1 + · · ·+ kt − t)K3 + Ks is Rmin(k1K2, . . . , ktK2) saturated.
Corollary
sat(n;Rmin(k1K2 + · · ·+ ktK2)) ≤ 3(k1 + · · ·+ kt − t)when n ≥ 3(k1 + · · ·+ kt − t)
Ferrara, Kim, Y.; 2014
sat(n;Rmin(k1K2 + · · ·+ ktK2)) = 3(k1 + · · ·+ kt − t)when n > 3(k1 + · · ·+ kt − t)
Construction is generally unique: vertex-disjoint triangles with isolates.
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 9 / 11
![Page 53: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/53.jpg)
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 10 / 11
![Page 54: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/54.jpg)
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 10 / 11
![Page 55: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/55.jpg)
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 10 / 11
![Page 56: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/56.jpg)
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 10 / 11
![Page 57: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/57.jpg)
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 10 / 11
![Page 58: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/58.jpg)
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 10 / 11
![Page 59: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/59.jpg)
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 10 / 11
![Page 60: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/60.jpg)
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 10 / 11
![Page 61: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/61.jpg)
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 10 / 11
![Page 62: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/62.jpg)
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 10 / 11
![Page 63: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/63.jpg)
Useful Observation
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturationof the forbidden subgraph Hi .
Corollary
If G is Rmin(H1, . . . ,Hk) saturated, then G = G1 ∪ · · · ∪ Gk , where Gi isHi saturated and all Gi share the same vertex set.
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 10 / 11
![Page 64: Saturation Number of Ramsey-Minimal Familieselyse/Yeager_Ramsey_Saturation_Detroit.… · Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2](https://reader033.vdocuments.us/reader033/viewer/2022051908/5ffc507d62adc530582896d1/html5/thumbnails/64.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.
- T. Szabo, On nearly regular co-critical graphs, Discrete Math. 160(1996) 279-281.
Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 11 / 11