A Ramsey Version of Graph Saturation
Mike Ferrara
Jaehoon Kim
Elyse Yeager
yeager2@illinois.edu
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
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
2/9
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
2/9
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
2/9
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph of G , but for every e ∈ G , H is a subgraph of G + e.
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
2/9
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph 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 smallest
number of edges over all n-vertex graphs that are H-saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
2/9
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph 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 smallest
number of edges over all n-vertex graphs that are H-saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
2/9
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph 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 smallest
number of edges over all n-vertex graphs that are H-saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
2/9
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph 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 smallest
number of edges over all n-vertex graphs that are H-saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
2/9
Graph Saturation
Definitions
Given a forbidden graph H, a graph G is H-saturated if H is not a
subgraph 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 smallest
number 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 no
member of F is a subgraph of G , but for every e ∈ G , some member of F
is a subgraph of G + e.
The saturation number sat(n;F) is the smallest number of edges over
all n-vertex graphs that are F-saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
2/9
Ramsey-Minimal Families
Definitions
A graph G is (H1 , . . . , Hk )-Ramsey minimal if G → (H1 , . . . , Hk ) but for
any e ∈ E (G ), G − e 6→ (H1 , . . . , Hk ).
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
3/9
Ramsey-Minimal Families
Definitions
A graph G is (H1 , . . . , Hk )-Ramsey minimal if G → (H1 , . . . , Hk ) but for
any e ∈ E (G ), G − e 6→ (H1 , . . . , Hk ).
Example: K6 → (K3 , K3 ), but K6 − e 6→ (K3 , K3 ).
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
3/9
Ramsey-Minimal Families
Definitions
A graph G is (H1 , . . . , Hk )-Ramsey minimal if G → (H1 , . . . , Hk ) but for
any e ∈ E (G ), G − e 6→ (H1 , . . . , Hk ).
Example: K6 → (K3 , K3 ), but K6 − e 6→ (K3 , K3 ).
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
3/9
Ramsey-Minimal Families
Definitions
A graph G is (H1 , . . . , Hk )-Ramsey minimal if G → (H1 , . . . , Hk ) but for
any e ∈ E (G ), G − e 6→ (H1 , . . . , Hk ).
Example: K6 → (K3 , K3 ), but K6 − e 6→ (K3 , K3 ).
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
3/9
Ramsey-Minimal Families
Definitions
A graph G is (H1 , . . . , Hk )-Ramsey minimal if G → (H1 , . . . , Hk ) but for
any e ∈ E (G ), G − e 6→ (H1 , . . . , Hk ).
Example: K6 → (K3 , K3 ), but K6 − e 6→ (K3 , K3 ).
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
3/9
Ramsey-Minimal Families
Definitions
A graph G is (H1 , . . . , Hk )-Ramsey minimal if G → (H1 , . . . , Hk ) but for
any 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
Saturation of Ramsey-Minimal Families
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
4/9
Saturation of Ramsey-Minimal Families
Rmin (H1 , . . . , Hk ) Saturation
A graph G is Rmin (H1 , . . . , Hk ) saturated if and only if:
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
4/9
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 )
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
4/9
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 ).
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
4/9
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
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
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
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
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
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
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
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
Ferrara-Kim-Yeager (UCD, UIUC)
n − (r − 2)
Ramsey Version of Saturation
24 October 2014
5/9
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
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
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
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
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
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
Kr −2 ∨ Ks 6→ (Kk1 , . . . , Kkt )
Ferrara-Kim-Yeager (UCD, UIUC)
n − (r − 2)
Kr −2 ∨ Ks + e → (Kk1 , . . . , Kkt )
Ramsey Version of Saturation
24 October 2014
5/9
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
Kr −2 ∨ Ks 6→ (Kk1 , . . . , Kkt )
Ferrara-Kim-Yeager (UCD, UIUC)
n − (r − 2)
Kr −2 ∨ Ks + e → (Kk1 , . . . , Kkt )
Ramsey Version of Saturation
24 October 2014
5/9
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
Kr −2 ∨ Ks 6→ (Kk1 , . . . , Kkt )
Ferrara-Kim-Yeager (UCD, UIUC)
n − (r − 2)
Kr −2 ∨ Ks + e → (Kk1 , . . . , Kkt )
Ramsey Version of Saturation
24 October 2014
5/9
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 )) ≤
Ferrara-Kim-Yeager (UCD, UIUC)
r −2
2
+ (r − 2)(n − r + 2) when n ≥ r
Ramsey Version of Saturation
24 October 2014
5/9
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 )) =
r −2
2
Ferrara-Kim-Yeager (UCD, UIUC)
n
2
n<r
+ (r − 2)(n − r + 2) n ≥ r
Ramsey Version of Saturation
24 October 2014
5/9
Hanson-Toft
Hanson-Toft Conjecture
sat(n; Rmin (Kk1 , . . . , Kkt )) =
r −2
2
Ferrara-Kim-Yeager (UCD, UIUC)
n
2
n<r
+ (r − 2)(n − r + 2) n ≥ r
Ramsey Version of Saturation
24 October 2014
6/9
Hanson-Toft
Hanson-Toft Conjecture
sat(n; Rmin (Kk1 , . . . , Kkt )) =
r −2
2
n
2
n<r
+ (r − 2)(n − r + 2) n ≥ r
Chen, Ferrara, Gould, Magnant, Schmitt; 2011
sat(n; Rmin (K3 , K3 )) =
Ferrara-Kim-Yeager (UCD, UIUC)
n
2
n<6=r
4n − 10 n ≥ 56
Ramsey Version of Saturation
24 October 2014
6/9
Hanson-Toft
Hanson-Toft Conjecture
sat(n; Rmin (Kk1 , . . . , Kkt )) =
r −2
2
n
2
n<r
+ (r − 2)(n − r + 2) n ≥ r
Chen, Ferrara, Gould, Magnant, Schmitt; 2011
sat(n; Rmin (K3 , K3 )) =
Ferrara-Kim-Yeager (UCD, UIUC)
n
2
n<6=r
4n − 10 n ≥ 56
Ramsey Version of Saturation
24 October 2014
6/9
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
7/9
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
(5K2 , 5K2 , 5K2 , 5K2 )
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
7/9
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
(5K2 , 5K2 , 5K2 , 5K2 )
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
7/9
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
(5K2 , 5K2 , 5K2 , 5K2 )
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
7/9
Matchings
Example
(k1 + · · · + kt − t)K3 + Ks is Rmin (k1 K2 , . . . , kt K2 ) saturated.
(5K2 , 5K2 , 5K2 , 5K2 )
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
7/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
make red-heavy
take red subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of the forbidden subgraph Hi .
Example: Forbidden graphs (3K2 , 3K2 ).
good coloring
make red-heavy
take red subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Ramsey Version of Saturation
24 October 2014
8/9
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of 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
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of 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
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of 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
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of 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
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of 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
Useful Observation: “Iterated Recoloring”
Ferrara, Kim, Y.; 2014
Looking (cleverly) at color i allows us to use results from graph saturation
of 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
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 saturation
numbers for matchings, Discrete Math. 322 (2014) 26-30.
- A. Galluccio, M. Simonovits, G. Simonyi, On the structure of
co-critical graphs, In: Graph Theory, Combinatorics and Algorithms,
Vol. 1, 2 (Kalamazoo, MI, 1992). Wiley-Intersci. Publ., Wiley, New
York, 1053-1071.
- D. Hanson, B. Toft, Edge-colored saturated graphs, J. Graph
Theory 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