results and open problems on saturated graphs of minimum sizemlavrov/seminar/2013-ferrara-b.pdf ·...

Results and Open Problems on Saturated Graphsof Minimum Size

Michael FerraraUniversity of Colorado Denver

UIUCJanuary 16, 2013

Mike Ferrara Results and Open Problems on Saturated Graphs of Minimum Size

F -saturated graphs

Definition

Given a family F of graphs, a graph G is F-saturated if Gcontains no member of F , but for any pair of nonadjacent verticesu and v in G , G + uv contains some member of F .

If F = {F}, we then say that G is F -saturated.


The Turan Problem

Problem (The Turan Problem)

Determine ex(n,F), the maximum number of edges in a graphthat contains no member of F as a subgraph.

Thus, Every F-free graph of order n with ex(n,F) edges isF-saturated.


The Turan Problem

Problem (The Turan Problem)

Determine ex(n,F), the maximum number of edges in a graphthat contains no member of F as a subgraph.

Thus, Every F-free graph of order n with ex(n,F) edges isF-saturated.

ex(n,F) is the maximum number of edges in an F-saturatedgraph.


sat(n,F)

Definition

The minimum number of edges in an F-saturated graph is denotedsat(n,F).

sat(n,F) is the saturation number or saturation function of F .


Erdos-Hajnal-Moon

Erdos, Hajnal and Moon introduced the sat function anddetermined sat(n,Kt) exactly.

Theorem (E-H-M 1964)

sat(n,Kt) = e(Kt−2 + Kn−t+2) =(

t−22

)

+ (t − 2)(n − t + 2).

Furthermore, Kt−2 + Kn−t+2 is the unique Kt-saturated graph ofminimum size.


Non-Monotonicity

Interestingly, sat(n,F) does not share many of the nice propertiesof ex(n,F).

ex(n,F ) ≤ ex(n + 1,F )


Non-Monotonicity


ex(n,F ) ≤ ex(n + 1,F )

F ′ ⊂ F ⇒ ex(n,F ′) ≤ ex(n,F )


Non-Monotonicity


ex(n,F ) ≤ ex(n + 1,F )

F ′ ⊂ F ⇒ ex(n,F ′) ≤ ex(n,F )

F1 ⊂ F2 ⇒ ex(n,F1) ≥ex(n,F2)


Non-Monotonicity


ex(n,F ) ≤ ex(n + 1,F )

F ′ ⊂ F ⇒ ex(n,F ′) ≤ ex(n,F )

F1 ⊂ F2 ⇒ ex(n,F1) ≥ex(n,F2)

sat(n,F ) 6≤ sat(n + 1,F )

F ′ ⊂ F 6⇒sat(n,F ′) ≤ sat(n,F )

F1 ⊂ F2 6⇒sat(n,F1) ≥ sat(n,F2)


sat(n, F ) 6≤ sat(n + 1, F )

Theorem (Kasonyi and Tuza 1986)

For n ≥ 4,

sat(n,P4) =

n

2 n even

n+32 n odd


F′ ⊂ F 6⇒ sat(n, F ′) ≤ sat(n, F )

By Erdos-Hajnal-Moon, sat(n,K4) = 2n − 3.


F′ ⊂ F 6⇒ sat(n, F ′) ≤ sat(n, F )

By Erdos-Hajnal-Moon, sat(n,K4) = 2n − 3.

However, sat(n,K4 + pendant) ≤ 32n.


Best known lower bound??

The best known general upper bound on sat(n,H) is due toKasonyi and Tuza in 1986.




A trivial lower bound:

sat(n,H) ≥ δ(H)−12 n.




A trivial lower bound:

sat(n,H) ≥ δ(H)−12 n.

Problem

For an arbitrary graph F determine a non-trivial lower bound onsat(n,F ).


(Some) Known Results

sat(n,H) has been studied for many classes of graphs.

K1,t and Pt (Kasonyi and Tuza 1986)

Matchings (Mader 1973, Kasonyi and Tuza 1986)

tKr and Kr ∪ Ks (Faudree, F, Gould and Jacobson 2009)

Trees (Faudree, Faudree, Gould, Jacobson 2009)

A Survey of Minimum Saturated Graphs (Faudree, Faudree,Schmitt - Dynamic Survey, EJC)


Theorem (Kasonyi and Tuza 1986;Faudree, Faudree, Gould,Jacobson 2009)

Let Sk denote a star of order k and S∗

kdenote the graph obtained

by subdividing one edge of Sk .

Then for any tree Tk 6= S∗

k−1,Sk of order k,

sat(n,S∗

k−1) < sat(n,Tk) < sat(n,Sk).


Theorem (Kasonyi and Tuza 1986;Faudree, Faudree, Gould,Jacobson 2009)

Let Sk denote a star of order k and S∗

kdenote the graph obtained

by subdividing one edge of Sk .

Then for any tree Tk 6= S∗

k−1,Sk of order k,

sat(n,S∗

k−1) < sat(n,Tk) < sat(n,Sk).

Question (FFGJ 2009)

Which trees of order k have the second largest/second smallestsaturation number?


Group Work

Problem (Group Work)

Determine sat(n,P5).


Group Work

Problem (Group Work)

Determine sat(n,P5).

Problem (Tuza)

Characterize those trees T such that, for n sufficiently large,

sat(n,T ) < n.


Theorem (FFGJ 2009)

Let T be a tree.

1 There exists a tree T1 such that

T ⊆ T1 and sat(n,T1) < n.


Theorem (FFGJ 2009)

Let T be a tree.

1 There exists a tree T1 such that

T ⊆ T1 and sat(n,T1) < n.

2 For any α > 0, there exists a tree T2 such that

T ⊆ T2 and sat(n,T2) ≥ αn.


Saturation Numbers for Cycles

By E-H-M,sat(n,C3) = sat(n,K3) = n − 1.




Theorem (Ollmann 1972)

For n ≥ 5,

sat(n,C4) =

⌈

3n − 5

2

⌉

.




Theorem (Ollmann 1972)

For n ≥ 5,

sat(n,C4) =

⌈

3n − 5

2

⌉

.

Theorem (Ya-Chen 2009)

For n ≥ 21,

sat(n,C5) =

⌈

10(n − 1)

7

⌉

.


At this time, no other exact values for cycles are known, but we dohave the following:

Theorem (Barefoot, Clark, Entringer, Porter, Szekely, Tuza 1996)

For all t ≥ 5 and n ≥ n(t),

(1 +c1t)n < sat(n,Ct) <

(

1 +c2t

)

n.


At this time, no other exact values for cycles are known, but we dohave the following:

Theorem (Barefoot, Clark, Entringer, Porter, Szekely, Tuza 1996)

For all t ≥ 5 and n ≥ n(t),

(1 +c1t)n < sat(n,Ct) <

(

1 +c2t

)

n.

Theorem (Furedi, Kim 2012)

For all t ≥ 7 and n ≥ 2t − 5,

(

1 +1

t + 2

)

n − 1 < sat(n,Ct) <

(

1 +1

t − 4

)

n +

(

t − 4

2

)

.


H-subdivisions

An H-subdivision is obtained from a (multi)graph H by replacingthe edges of H with internally disjoint paths.

We let S(H) denote the family of H-subdivisions.


sat for Subdivisions

Every subdivision of a cycle is a (longer) cycle, so

S(Ct) = {Ck : k ≥ t}.


sat for Subdivisions

Every subdivision of a cycle is a (longer) cycle, so

S(Ct) = {Ck : k ≥ t}.

Thus, sat(n,S(Ct)) is the minimum number of edges in a graph Gwith

circ(G ) < t

such thatcirc(G + uv) ≥ t.


Short Cycles

sat(n,S(C3)) = sat(n,C3) = n − 1.


Short Cycles

sat(n,S(C3)) = sat(n,C3) = n − 1.

Theorem (F, Jacobson, Milans, Tennenhouse, Wenger 2012)

For n ≥ 1, sat(n,S(C4)) = n + ⌊n−34 ⌋.


C5

Theorem (Ya-Chen 2009)

For n ≥ 21,

sat(n,C5) =

⌈

10(n − 1)

7

⌉

.

Theorem (F, Jacobson, Milans, Tennenhouse, Wenger 2012)

For n ≥ 5,

sat(n,S(C5)) =

⌈

10(n − 1)

7

⌉

.


Asymptotics

Theorem (F, Jacobson, Milans, Wenger, Tennenhouse 2012)

For t ≥ 6 and n ≥ n(t), there exists an absolute constant c suchthat

5n

4≤ sat(n,S(Ct)) ≤ (

5

4+

c

t)n.

In particular if t ≥ 36, then c = 8 will suffice.


Asymptotics

Theorem (F, Jacobson, Milans, Wenger, Tennenhouse 2012)

For t ≥ 6 and n ≥ n(t), there exists an absolute constant c suchthat

5n

4≤ sat(n,S(Ct)) ≤ (

5

4+

c

t)n.

In particular if t ≥ 36, then c = 8 will suffice.

Recall that by Furedi-Kim,

sat(n,Ct) ≈

(

1 +1

t

)

n.

Our result implies that “long” cycles are an essential part ofCt -saturated graphs with minimum size.


An Interesting Result

Theorem (Clark, Etringer, Shapiro 1983/1992)

For n ≥ 54,

sat(n,Cn) =


An Interesting Result

Theorem (Clark, Etringer, Shapiro 1983/1992)

For n ≥ 54,

sat(n,Cn) =

⌈

3n

2

⌉

.


Upper Bound Construction

Definition

A snark is a bridgeless, cubic graph with girth at least five andedge chromatic number four.



Definition


We modify the snark-based construction of Barefoot et al. toconstruct S(Ct)-saturated graphs for sporadic small t and allt ≥ 36.



Definition


We modify a snark-based construction of Barefoot et al. toconstruct S(Ct)-saturated graphs for sporadic small t and allt ≥ 36.


Problems

Conjecture

There exist absolute constants c1 and c2 such that for all t and n,

(5

4+

c1t)n < sat(n,S(Ct)) <

(

5

4+

c2t

)

n.


Subdivisions of Cliques

Turning our attention to subdivided cliques:

sat(S(K3), n) = sat(S(C3), n) = n − 1.

Proposition

G is S(K4)-saturated iff G is a 2-tree.

Consequently,

sat(n,S(K4)) = ex(n,S(K4)) = sat(n,K4) = 2n − 3.


Subdivisions of Cliques

Theorem (Erdos-Hajnal-Moon 1964)

sat(n,Kt) = e(Kt−2 ∨ Kn−t+2) =(

t−22

)

+ (t − 2)(n − t + 2).

Theorem (F, Jacobson, Milans, Wenger, Tennenhouse, 2012)

Let t ≥ 5 and n = d(t − 1) + r for d ≥ 2 and 0 ≤ r ≤ t − 2. Then

sat(n,S(Kt)) ≤

(

t − 2

2+ o(1)

)

n.


Construction - t odd


t = 5


If t = 4d + r


t even


The extremal function for S(K5)

In 1998, Mader determined the extremal number of S(K5),affirming a 1964 conjecture of Dirac.

Theorem (Mader 1998)

ex(n,S(K5)) = 3n − 6.


sat(n,S(K5))

Theorem (F, Jacobson, Milans, Wenger, Tennenhouse, 2012)

For n ≥ 10,

sat(n,S(K5)) =

⌈

3n + 4

2

⌉

.


K5-minors

Let M(K5) denote the family of graphs that have a K5-minor.

Theorem (Wagner 1937)

If G is an M(K5)-saturated graph of order at least 4, then G canbe obtained from maximally planar graphs and copies of the graphW by pasting along edges and triangles.


Non-Monotonicity

Wagner’s Theorem implies that

sat(n,M(K5)) ≈11

6n.

Therefore S(K5) ⊆ M(K5), but

⌈

3n + 4

2

⌉

= sat(n,S(K5)) ≤ sat(n,M(K5)) ≈11

6n.


Non-Monotonicity


Problems

Question (Hard!)

Is there some absolute constant c such that for all t ≥ 3,

sat(n,S(Kt)) ≤ cn?


Problems

Question (Hard!)



Question (Start Here!)

Determine sat(n,S(K6)) for n sufficiently large.


Problems

Question (Hard!)



Question (Start Here!)

Determine sat(n,S(K6)) for n sufficiently large.

Problem

Investigatesat(n,M(Kt))

for n ≥ 6.


→

Definition

Given graphs G ,H1, . . . ,Hk , we write

G → (H1,H2, . . . ,Hk)

if every k-edge coloring of G contains a copy of Hi in color i forsome i .


→

Definition

Given graphs G ,H1, . . . ,Hk , we write

G → (H1,H2, . . . ,Hk)

if every k-edge coloring of G contains a copy of Hi in color i forsome i .

Therefore, the classical Ramsey number r(H1, . . . ,Hk) is thesmallest n such that

Kn−1 6 → (H1, . . . ,Hk),

andKn → (H1, . . . ,Hk).


A Problem of Hanson and Toft

In 1987, Hanson and Toft posed the following problem:

Problem

Let t1, . . . , tk be positive integers. Determine the minimumnumber of edges in a graph G of order n such that:

1 there is a k-edge coloring of G with no monochromatic Ktiin

color i for any i , and




Problem

Let t1, . . . , tk be positive integers. Determine the minimumnumber of edges in a graph G of order n such that:

1 there is a k-edge coloring of G with no monochromatic Ktiin

color i for any i , and

2 for any uv ∈ G, every k-edge-coloring of G + uv contains amonochromatic copy of Kti

in color i for some i .




Problem (Alternate Formulation)

Determine the minimum number of edges in a graph G of order nsuch that

G 6 → (Kt1 , . . . ,Ktk)

but for any uv ∈ G ,

G + uv → (Kt1 , . . . ,Ktk).


(H1, . . . ,Hk)-Ramsey Minimality

Definition

Given graphs G ,H1, . . . ,Hk , we say that G is(H1, . . . ,Hk)-Ramsey-minimal if G → (H1, . . . ,Hk), butG ′ 6 → (H1, . . . ,Hk) for any proper subgraph G ′ of G .

Example: C5 is (P3,P3)-Ramsey-minimal.


An Important Observation:

Observation

G → (H1, . . . ,Hk) if and only if G contains an(H1, . . . ,Hk)-Ramsey-minimal subgraph.



Observation

G → (H1, . . . ,Hk) if and only if G contains an(H1, . . . ,Hk)-Ramsey-minimal subgraph.

Problem (Hanson and Toft)

Determine the minimum number of edges in a graph G such that

G 6 → (Kt1 , . . . ,Ktk)

but for any uv ∈ G ,

G + uv → (Kt1 , . . . ,Ktk).



Problem (Hanson and Toft)

Determine sat(n,Rmin(Kt1 , . . . ,Ktk)).


First Steps

Conjecture (Hanson and Toft 1987)

Let r = r(Kt1 , . . . ,Ktk). Then

sat(n,Rmin(Kt1 , . . . ,Ktk)) = sat(n,Kr ).

The first step to affirm the conjecture would be to demonstrate anappropriate red/blue coloring of a minimal Kr -saturated graph.


sat(n,Rmin(K3,K3))

Recall that r(K3,K3) = 6.

We show that sat(n,Rmin(K3,K3)) ≤ sat(n,K6) = 4n − 10.


sat(n,Rmin(K3,K3))

Recall that r(K3,K3) = 6.

We show that sat(n,Rmin(K3,K3)) ≤ sat(n,K6) = 4n − 10.

To do so, we demonstrate a red/blue coloring of K4 + Kn−4 withno monochromatic K3.


K4 + Kn−4 6 → (K3,K3)


t = ℓ = 3

This example, and similar constructions, are the motivation for theconjecture.


t = ℓ = 3

This example, and similar constructions, are the motivation for theconjecture.

We have verified the conjecture for t = ℓ = 3, which represents thefirst nontrivial progress towards Hanson-Toft.

Theorem (Chen, F, Gould, Magnant, Schmitt 2011)

For n ≥ 56,

sat(n,Rmin(K3,K3)) = sat(n,K6) = 4n − 10.


Problems

Problem (Hard!)

Determine sat(n,Rmin(Km,Kn)) for all m, n ≥ 3.

Problem (Start Here!)

Determine sat(n,Rmin(K3,K4)).


Problems

Theorem (CFGMS 2011)

sat(n,Rmin(K3,P3)) =

⌊

5n

2

⌋

− 5.

Problem

Determine sat(n,Rmin(G1,G2)) for “interesting” G1 and G2.


results and open problems on saturated graphs of minimum sizemlavrov/seminar/2013-ferrara-b.pdf ·...

Documents