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### Extremal stable graphs

Illés Horváth1 Gyula Y. Katona2

1Department of Stochastics

Budapest University of Technology and Economics

2Department of Computer Science and Information Theory Budapest University of Technology and Economics

3 June, 2009

(2)

### Outline

1 Introduction

2 Main result

3 Outline of proof

4 Application

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 2 / 26

(3)

### Introduction

Question

LetΠbe a graph property so that if G1∈Πand G1is a subgraph of G2, then G2∈Π.(i. e. being non-Πis a hereditary graph property.)

What is the minimum number of edges in a graph G∈Πon n vertices if removing any k edges (or vertices) from the graph still preservesΠ? Examples:

What is the minimum number of edges in ak-connected or k-edge connected graph?

What is the miminum number of edges in hypo-hamiltonian graph? What is the mininum number of edges in graph that is still

Hamiltonian after removingk edges (or vertices)?

(4)

### Introduction

Question

LetΠbe a graph property so that if G1∈Πand G1is a subgraph of G2, then G2∈Π.(i. e. being non-Πis a hereditary graph property.) What is the minimum number of edges in a graph G∈Πon n vertices if removing any k edges (or vertices) from the graph still preservesΠ?

Examples:

What is the minimum number of edges in ak-connected or k-edge connected graph?

What is the miminum number of edges in hypo-hamiltonian graph? What is the mininum number of edges in graph that is still

Hamiltonian after removingk edges (or vertices)?

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 3 / 26

(5)

### Introduction

Question

LetΠbe a graph property so that if G1∈Πand G1is a subgraph of G2, then G2∈Π.(i. e. being non-Πis a hereditary graph property.) What is the minimum number of edges in a graph G∈Πon n vertices if removing any k edges (or vertices) from the graph still preservesΠ?

Examples:

What is the minimum number of edges in ak-connected or k-edge connected graph?

What is the miminum number of edges in hypo-hamiltonian graph? What is the mininum number of edges in graph that is still

Hamiltonian after removingk edges (or vertices)?

(6)

### Introduction

Question

LetΠbe a graph property so that if G1∈Πand G1is a subgraph of G2, then G2∈Π.(i. e. being non-Πis a hereditary graph property.) What is the minimum number of edges in a graph G∈Πon n vertices if removing any k edges (or vertices) from the graph still preservesΠ?

Examples:

What is the minimum number of edges in ak-connected or k-edge connected graph?

What is the miminum number of edges in hypo-hamiltonian graph?

What is the mininum number of edges in graph that is still Hamiltonian after removingk edges (or vertices)?

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 3 / 26

(7)

### Introduction

Question

LetΠbe a graph property so that if G1∈Πand G1is a subgraph of G2, then G2∈Π.(i. e. being non-Πis a hereditary graph property.) What is the minimum number of edges in a graph G∈Πon n vertices if removing any k edges (or vertices) from the graph still preservesΠ?

Examples:

What is the minimum number of edges in ak-connected or k-edge connected graph?

What is the miminum number of edges in hypo-hamiltonian graph?

What is the mininum number of edges in graph that is still Hamiltonian after removingk edges (or vertices)?

(8)

### Notations and definitions

We concentrate on the problem whereΠis the property thatG contains a given fixed subgraphH.

This clearly satisfies the assumption onΠ. We only consider simple, undirected graphs.

Definition (Stability)

Let H be a fixed graph. If the graph G has the property that removing any k edges of G, the resulting graph still contains(not necessarily spans)a subgraph isomorphic with H, then we say that G isk -stable with regard to H.

Definition

BySH(k)we denote the minimum number of edges in any k -stable graph.

Note that there is no “in a graph withnvertices” in the definition.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 4 / 26

(9)

### Notations and definitions

We concentrate on the problem whereΠis the property thatG contains a given fixed subgraphH.

This clearly satisfies the assumption onΠ.

We only consider simple, undirected graphs.

Definition (Stability)

Let H be a fixed graph. If the graph G has the property that removing any k edges of G, the resulting graph still contains(not necessarily spans)a subgraph isomorphic with H, then we say that G isk -stable with regard to H.

Definition

BySH(k)we denote the minimum number of edges in any k -stable graph.

Note that there is no “in a graph withnvertices” in the definition.

(10)

### Notations and definitions

We concentrate on the problem whereΠis the property thatG contains a given fixed subgraphH.

This clearly satisfies the assumption onΠ.

We only consider simple, undirected graphs.

Definition (Stability)

Let H be a fixed graph. If the graph G has the property that removing any k edges of G, the resulting graph still contains(not necessarily spans)a subgraph isomorphic with H, then we say that G isk -stable with regard to H.

Definition

BySH(k)we denote the minimum number of edges in any k -stable graph.

Note that there is no “in a graph withnvertices” in the definition.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 4 / 26

(11)

### Notations and definitions

We concentrate on the problem whereΠis the property thatG contains a given fixed subgraphH.

This clearly satisfies the assumption onΠ.

We only consider simple, undirected graphs.

Definition (Stability)

Let H be a fixed graph. If the graph G has the property that removing any k edges of G, the resulting graph still contains(not necessarily spans)a subgraph isomorphic with H, then we say that G isk -stable with regard to H.

Definition

BySH(k)we denote the minimum number of edges in any k -stable graph.

Note that there is no “in a graph withnvertices” in the definition.

(12)

### Notations and definitions

We concentrate on the problem whereΠis the property thatG contains a given fixed subgraphH.

This clearly satisfies the assumption onΠ.

We only consider simple, undirected graphs.

Definition (Stability)

Let H be a fixed graph. If the graph G has the property that removing any k edges of G, the resulting graph still contains(not necessarily spans)a subgraph isomorphic with H, then we say that G isk -stable with regard to H.

Definition

BySH(k)we denote the minimum number of edges in any k -stable graph.

Note that there is no “in a graph withnvertices” in the definition.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 4 / 26

(13)

### Notations and definitions

We concentrate on the problem whereΠis the property thatG contains a given fixed subgraphH.

This clearly satisfies the assumption onΠ.

We only consider simple, undirected graphs.

Definition (Stability)

Let H be a fixed graph. If the graph G has the property that removing any k edges of G, the resulting graph still contains(not necessarily spans)a subgraph isomorphic with H, then we say that G isk -stable with regard to H.

Definition

BySH(k)we denote the minimum number of edges in any k -stable graph.

(14)

### A trivial case: H = P

2

Proposition SP2(k) =k +1

Extremal graph: Any graph withk +1 edges. Lemma

For any graph H, we have SH(k)≥k +|E(H)|. Proof.

Otherwise there aren’t enough edges to formH.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 5 / 26

(15)

### A trivial case: H = P

2

Proposition SP2(k) =k +1

Extremal graph: Any graph withk +1 edges.

Lemma

For any graph H, we have SH(k)≥k +|E(H)|. Proof.

Otherwise there aren’t enough edges to formH.

(16)

### A trivial case: H = P

2

Proposition SP2(k) =k +1

Extremal graph: Any graph withk +1 edges.

Lemma

For any graph H, we have SH(k)≥k +|E(H)|.

Proof.

Otherwise there aren’t enough edges to formH.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 5 / 26

(17)

### A trivial case: H = P

2

Proposition SP2(k) =k +1

Extremal graph: Any graph withk +1 edges.

Lemma

For any graph H, we have SH(k)≥k +|E(H)|.

Proof.

Otherwise there aren’t enough edges to formH.

(18)

### An easy case: H = P

3

Proposition

SP3(k) =k +2=k+|E(P3)|

Extremal graph: A star withk +2 edges.

1 2 3 4 k+1 k+2

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 6 / 26

(19)

### An easy case: H = P

3

Proposition

SP3(k) =k +2=k+|E(P3)|

Extremal graph: A star withk +2 edges.

1 2 3 4 k+1 k+2

(20)

### An other easy case: H = 2P

2

Proposition

S2P2(k) =k+2=k +|E(2P2)|

Extremal graph: k +2 independent edges.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 7 / 26

(21)

### An other easy case: H = 2P

2

Proposition

S2P2(k) =k+2=k +|E(2P2)|

Extremal graph: k +2 independent edges.

(22)

### Linear bounds

Proposition Let H be fixed.

(a) S(k)≥k+|E(H)|.

(b) S(k)≤(|V(H)|+1)k if k is large enough.

Proof. (a) is trivial,

(b) is a consequence of Turán’s theorem.

For a fixedH graphs we are interested in the exact value ofS(k)and also the extremal graphs.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 8 / 26

(23)

### Linear bounds

Proposition Let H be fixed.

(a) S(k)≥k+|E(H)|.

(b) S(k)≤(|V(H)|+1)k if k is large enough.

Proof.

(a) is trivial,

(b) is a consequence of Turán’s theorem.

For a fixedH graphs we are interested in the exact value ofS(k)and also the extremal graphs.

(24)

### Linear bounds

Proposition Let H be fixed.

(a) S(k)≥k+|E(H)|.

(b) S(k)≤(|V(H)|+1)k if k is large enough.

Proof.

(a) is trivial,

(b) is a consequence of Turán’s theorem.

For a fixedH graphs we are interested in the exact value ofS(k)and also the extremal graphs.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 8 / 26

(25)

### Linear bounds

Proposition Let H be fixed.

(a) S(k)≥k+|E(H)|.

(b) S(k)≤(|V(H)|+1)k if k is large enough.

Proof.

(a) is trivial,

(b) is a consequence of Turán’s theorem.

For a fixedH graphs we are interested in the exact value ofS(k)and also the extremal graphs.

(26)

### Outline

1 Introduction

2 Main result

3 Outline of proof

4 Application

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 9 / 26

(27)

### Main result: H = P

4

Theorem

S(1) =4, and for k ≥2,

S(k) =k+

&r 2k+9

4 +3 2 '

.

The above formula is equivalent with the following:

Theorem

S(1) =4,S(2) =6, and if k ≥3, S(k) =

S(k−1) +2 ifk = 2`

for some integer` S(k−1) +1 otherwise

(28)

### Main result: H = P

4

Theorem

S(1) =4, and for k ≥2,

S(k) =k+

&r 2k+9

4 +3 2 '

.

The above formula is equivalent with the following:

Theorem

S(1) =4,S(2) =6, and if k ≥3, S(k) =

S(k−1) +2 ifk = 2`

for some integer` S(k−1) +1 otherwise

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 10 / 26

(29)

### Outline

1 Introduction

2 Main result

3 Outline of proof

4 Application

(30)

### Covering with triangles and stars

Proposition

If G does not contain P4as a subgraph, then every component of G is a triangle or a star.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 12 / 26

(31)

### Covering with triangles and stars

Proposition

If G is a graph witheedges onnvertices, then

G is k -stable ⇐⇒ the vertices of G cannot be covered byk +n−e stars and any number of triangles.

Proof.

Gis notk-stable if there is a subgraph withe−k edges ofGsuch that it does not containP4.

That subgraph is a union of triangles and stars, and the number of stars isn−(e−k).

(In triangles, the number of edges is equal to the number of vertices, while in a star, the number of edges is 1 less, so we “lose” an edge for every star.)

(32)

### Covering with triangles and stars

Proposition

If G is a graph witheedges onnvertices, then

G is k -stable ⇐⇒ the vertices of G cannot be covered byk +n−e stars and any number of triangles.

Proof.

Gis notk-stable if there is a subgraph withe−k edges ofGsuch that it does not containP4.

That subgraph is a union of triangles and stars, and the number of stars isn−(e−k).

(In triangles, the number of edges is equal to the number of vertices, while in a star, the number of edges is 1 less, so we “lose” an edge for every star.)

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 13 / 26

(33)

### Covering with triangles and stars

Proposition

If G is a graph witheedges onnvertices, then

G is k -stable ⇐⇒ the vertices of G cannot be covered byk +n−e stars and any number of triangles.

Proof.

Gis notk-stable if there is a subgraph withe−k edges ofGsuch that it does not containP4.

That subgraph is a union of triangles and stars, and the number of stars isn−(e−k).

(In triangles, the number of edges is equal to the number of vertices, while in a star, the number of edges is 1 less, so we “lose” an edge for every star.)

(34)

### Covering with triangles and stars

Proposition

If G is a graph witheedges onnvertices, then

G is k -stable ⇐⇒ the vertices of G cannot be covered byk +n−e stars and any number of triangles.

Proof.

Gis notk-stable if there is a subgraph withe−k edges ofGsuch that it does not containP4.

That subgraph is a union of triangles and stars, and the number of stars isn−(e−k).

(In triangles, the number of edges is equal to the number of vertices, while in a star, the number of edges is 1 less, so we “lose” an edge for every star.)

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 13 / 26

(35)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =3,S(k) =8

e= 7 n= 5

k+n−e= 1

(36)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =3,S(k) =8

e= 7 n= 5

k+n−e= 1

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 14 / 26

(37)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =3,S(k) =8

e= 7 n= 5

k+n−e= 1

(38)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =3,S(k) =8

e= 7 n= 5

k+n−e= 1

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 14 / 26

(39)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =3,S(k) =8

e= 7 n= 5

k+n−e= 1

(40)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =3,S(k) =8

e= 7 n= 5

k+n−e= 1

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 14 / 26

(41)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =3,S(k) =8

e= 7 n= 5

k+n−e= 1

(42)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =3,S(k) =8

## k + n − e = 2

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 15 / 26

(43)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =3,S(k) =8

## k + n − e = 2

(44)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =3,S(k) =8

## k + n − e = 2

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 15 / 26

(45)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =3,S(k) =8

e= 7 n= 8

k+n−e= 4

(46)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =3,S(k) =8

e= 7 n= 8

k+n−e= 4

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 16 / 26

(47)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =3,S(k) =8

e= 7 n= 8

k+n−e= 4

(48)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =6,S(k) =12

## k + n − e = 2 e = 11

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 17 / 26

(49)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =6,S(k) =12

## e = 11

(50)

### Examples for the lower bound

We need to show that any graph with<S(k)edges is notk-stable

⇐⇒ the vertices cannot be covered byk +n−estars and any number of triangles.

k =6,S(k) =12

## k + n − e = 2 e = 11

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 17 / 26

(51)

### Lower bound

Definition

Given a graph G with e≥5edges on n vertices, let`be the largest integer such that e≥ `−12

+1(that is,`is the smallest possible number of vertices that can fit e edges), and let s =n−`.

s ≥0because of the definition of`; s measures how ’spread-out’ G is.

Theorem (Lower bound)

If the graph G has e≥5edges, then G can be covered by s+1stars and any number of triangles.

S(k)≥k + q

2k +94+32

follows directly from the above.

(52)

### Lower bound

Definition

Given a graph G with e≥5edges on n vertices, let`be the largest integer such that e≥ `−12

+1(that is,`is the smallest possible number of vertices that can fit e edges), and let s =n−`.

s ≥0because of the definition of`; s measures how ’spread-out’ G is.

Theorem (Lower bound)

If the graph G has e≥5edges, then G can be covered by s+1stars and any number of triangles.

S(k)≥k + q

2k +94+32

follows directly from the above.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 18 / 26

(53)

### Lower bound

Definition

Given a graph G with e≥5edges on n vertices, let`be the largest integer such that e≥ `−12

+1(that is,`is the smallest possible number of vertices that can fit e edges), and let s =n−`.

s ≥0because of the definition of`; s measures how ’spread-out’ G is.

Theorem (Lower bound)

If the graph G has e≥5edges, then G can be covered by s+1stars and any number of triangles.

S(k)≥k + q

2k +94+32

follows directly from the above.

(54)

### Lower bound - cases

For different values ofs, the methods are different.

Lemma (s =0 ors =1)

The vertices can be covered by s+1stars andat most 1 triangle. The proof is long but elementary.

Lemma (s ≥2)

The vertices can be covered by s+1stars. No triangles are needed.

If only stars are used, then the centers of the stars forms a ”dominating vertex set”.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 19 / 26

(55)

### Lower bound - cases

For different values ofs, the methods are different.

Lemma (s =0 ors =1)

The vertices can be covered by s+1stars andat most 1 triangle.

The proof is long but elementary.

Lemma (s ≥2)

The vertices can be covered by s+1stars. No triangles are needed.

If only stars are used, then the centers of the stars forms a ”dominating vertex set”.

(56)

### Lower bound - cases

For different values ofs, the methods are different.

Lemma (s =0 ors =1)

The vertices can be covered by s+1stars andat most 1 triangle.

The proof is long but elementary.

Lemma (s ≥2)

The vertices can be covered by s+1stars. No triangles are needed.

If only stars are used, then the centers of the stars forms a ”dominating vertex set”.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 19 / 26

(57)

### Lower bound - cases

For different values ofs, the methods are different.

Lemma (s =0 ors =1)

The vertices can be covered by s+1stars andat most 1 triangle.

The proof is long but elementary.

Lemma (s ≥2)

The vertices can be covered by s+1stars.

No triangles are needed.

If only stars are used, then the centers of the stars forms a ”dominating vertex set”.

(58)

### Lower bound - cases

For different values ofs, the methods are different.

Lemma (s =0 ors =1)

The vertices can be covered by s+1stars andat most 1 triangle.

The proof is long but elementary.

Lemma (s ≥2)

The vertices can be covered by s+1stars.

No triangles are needed.

If only stars are used, then the centers of the stars forms a ”dominating vertex set”.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 19 / 26

(59)

### Lower bound - cases

For different values ofs, the methods are different.

Lemma (s =0 ors =1)

The vertices can be covered by s+1stars andat most 1 triangle.

The proof is long but elementary.

Lemma (s ≥2)

The vertices can be covered by s+1stars.

No triangles are needed.

If only stars are used, then the centers of the stars forms a ”dominating vertex set”.

(60)

### Lower bound - cases

The proof uses the following theorem:

Theorem (Vizing, 1965)

If G is a connected graph onnvertices andeedges, then the vertices can be dominated by a set of size

β(G)≤

1+2n−√ 8e+1 2

if e≤ (n−2)(n−3)2 .

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 20 / 26

(61)

### Upper bound

Theorem (Upper bound)

S(1) =4, and for k ≥2, S(k)≤k+ q

2k+94+32

.

Rephrased for coverings:

Theorem

For every k ≥2, there exists a graph G with e=k+ q

2k+94+ 32

edges that is k -stable, that is, it cannot be covered by s =k+n−e stars and any number of triangles.

(62)

### Upper bound

Theorem (Upper bound)

S(1) =4, and for k ≥2, S(k)≤k+ q

2k+94+32

. Rephrased for coverings:

Theorem

For every k ≥2, there exists a graph G with e=k+ q

2k+94+ 32

edges that is k -stable, that is, it cannot be covered by s =k+n−e stars and any number of triangles.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 21 / 26

(63)

### Upper bound

Proof.

Let`be the unique integer for which `−22

≤k ≤ `−12

−1.

There are 2 types of constructions:

1 If 3|`, then an almost complete graph,

2 If 3|`, then a complete graph with pendant edges.

(64)

### Upper bound

Proof.

Let`be the unique integer for which `−22

≤k ≤ `−12

−1.

There are 2 types of constructions:

1 If 3|`, then an almost complete graph,

2 If 3|`, then a complete graph with pendant edges.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 22 / 26

(65)

### Upper bound

Proof.

Let`be the unique integer for which `−22

≤k ≤ `−12

−1.

There are 2 types of constructions:

1 If 3|`, then an almost complete graph,

2 If 3|`, then a complete graph with pendant edges.

(66)

### Constructions

k S(k) extremal graph

1 4

1 6

1 6

1 6

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 23 / 26

(67)

### Outline

1 Introduction

2 Main result

3 Outline of proof

4 Application

(68)

### Application

Definition

A v1,v2, . . . ,vnpermutation of the vertices of an r -regular hypergraph is a Hamiltonian chain if any r (cyclically) consecutive vertices form an edge.

Definition

A hypergraph is k -edge-hamiltonian if it has the property that removing any k edges, the resulting hypergraph still contains a Hamiltonian chain.

Theorem (Frankl, Katona)

For every 3-regular k -edge-hamiltonian hypergraph with h edges on n vertices,

h≥ S(k) 3 n.

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 25 / 26

(69)

### Application

Definition

A v1,v2, . . . ,vnpermutation of the vertices of an r -regular hypergraph is a Hamiltonian chain if any r (cyclically) consecutive vertices form an edge.

Definition

A hypergraph is k -edge-hamiltonian if it has the property that removing any k edges, the resulting hypergraph still contains a Hamiltonian chain.

Theorem (Frankl, Katona)

For every 3-regular k -edge-hamiltonian hypergraph with h edges on n vertices,

h≥ S(k) n.

(70)

### Bibliography

P. FRANKL, G. Y. KATONA,

Extremalk-edge-hamiltonian hypergraphs,

Discrete Mathematics (2008)308, pp. 1415–1424 M. PAOLI, W. W. WONG, C. K. WONG,

Minimumk-Hamiltonian graphs. II.,

J. Graph Theory (1986)10, no. 1, pp. 79–95 D. RAUTENBACH,

A linear Vizing-like relation between the size and the domination number of a graph,

J. Graph Theory (1999)31, no. 4, pp. 297–302 V. G. VIZING,

An estimate of the external stability number of a graph, Dokl. Akad. Nauk SSSR (1965)52, pp. 729–731

Gyula Y. Katona (Hungary) Extremal stable graphs CTW09 26 / 26

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