Saturation Number of Ramsey-Minimal Families Mike Ferrara 1 Jaehoon Kim 2 Elyse Yeager 2 1 University of Colorado-Denver 2 University of Illinois at Urbana-Champaign [email protected] MIGHTY University of Detroit Mercy 29 March 2014 Ferrara-Kim-Yeager (UCD, UIUC) Saturation of Ramsey-Minimal Families 29 March 2014 1 / 11
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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