Extremal stable graphs

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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

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Outline

1 Introduction

2 Main result

3 Outline of proof

4 Application

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

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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)?

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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:

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Introduction

Question

Examples:

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Introduction

Question

Examples:

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)?

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Introduction

Question

Examples:

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)?

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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

ByS_H(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.

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Notations and definitions

This clearly satisfies the assumption onΠ.

We only consider simple, undirected graphs.

Definition

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Notations and definitions

Definition

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Notations and definitions

Definition

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Notations and definitions

Definition

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Notations and definitions

Definition

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A trivial case: H = P

₂

Proposition S_P₂(k) =k +1

Extremal graph: Any graph withk +1 edges. Lemma

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

Otherwise there aren’t enough edges to formH.

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A trivial case: H = P

₂

Extremal graph: Any graph withk +1 edges.

Lemma

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

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A trivial case: H = P

₂

Lemma

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

Proof.

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A trivial case: H = P

₂

Lemma

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

Proof.

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An easy case: H = P

₃

Proposition

S_P₃(k) =k +2=k+|E(P₃)|

Extremal graph: A star withk +2 edges.

1 2 3 4 k+1 k+2

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An easy case: H = P

₃

Proposition

S_P₃(k) =k +2=k+|E(P₃)|

Extremal graph: A star withk +2 edges.

1 2 3 4 k+1 k+2

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An other easy case: H = 2P

₂

Proposition

S_2P₂(k) =k+2=k +|E(2P₂)|

Extremal graph: k +2 independent edges.

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An other easy case: H = 2P

₂

Proposition

S_2P₂(k) =k+2=k +|E(2P₂)|

Extremal graph: k +2 independent edges.

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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.

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Linear bounds

(a) S(k)≥k+|E(H)|.

Proof.

(a) is trivial,

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Linear bounds

(a) S(k)≥k+|E(H)|.

Proof.

(a) is trivial,

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Linear bounds

(a) S(k)≥k+|E(H)|.

Proof.

(a) is trivial,

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Outline

1 Introduction

2 Main result

3 Outline of proof

4 Application

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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

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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

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Outline

1 Introduction

2 Main result

3 Outline of proof

4 Application

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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.

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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.)

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Covering with triangles and stars

Proposition

Proof.

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Covering with triangles and stars

Proposition

Proof.

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Covering with triangles and stars

Proposition

Proof.

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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

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Examples for the lower bound

k =3,S(k) =8

e= 7 n= 5

k+n−e= 1

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Examples for the lower bound

k =3,S(k) =8

e= 7 n= 5

k+n−e= 1

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Examples for the lower bound

k =3,S(k) =8

e= 7 n= 5

k+n−e= 1

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Examples for the lower bound

k =3,S(k) =8

e= 7 n= 5

k+n−e= 1

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Examples for the lower bound

k =3,S(k) =8

e= 7 n= 5

k+n−e= 1

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Examples for the lower bound

k =3,S(k) =8

e= 7 n= 5

k+n−e= 1

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Examples for the lower bound

k =3,S(k) =8

e = 7 n = 6

k + n − e = 2

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Examples for the lower bound

k =3,S(k) =8

e = 7 n = 6

k + n − e = 2

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Examples for the lower bound

k =3,S(k) =8

e = 7 n = 6

k + n − e = 2

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Examples for the lower bound

k =3,S(k) =8

e= 7 n= 8

k+n−e= 4

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Examples for the lower bound

k =3,S(k) =8

e= 7 n= 8

k+n−e= 4

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Examples for the lower bound

k =3,S(k) =8

e= 7 n= 8

k+n−e= 4

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Examples for the lower bound

k =6,S(k) =12

n = 7

k + n − e = 2 e = 11

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Examples for the lower bound

k =6,S(k) =12

n = 7

k + n − e = 2

e = 11

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Examples for the lower bound

k =6,S(k) =12

n = 7

k + n − e = 2 e = 11

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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.

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Lower bound

Definition

S(k)≥k + q

2k +⁹₄+³₂

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Lower bound

Definition

S(k)≥k + q

2k +⁹₄+³₂

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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”.

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Lower bound - cases

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)

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Lower bound - cases

Lemma (s =0 ors =1)

Lemma (s ≥2)

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Lower bound - cases

Lemma (s =0 ors =1)

Lemma (s ≥2)

The vertices can be covered by s+1stars.

No triangles are needed.

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Lower bound - cases

Lemma (s =0 ors =1)

Lemma (s ≥2)

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Lower bound - cases

Lemma (s =0 ors =1)

Lemma (s ≥2)

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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 .

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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.

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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.

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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.

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Upper bound

Proof.

≤k ≤ ^`−1₂

−1.

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Upper bound

Proof.

≤k ≤ ^`−1₂

−1.

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Constructions

k S(k) extremal graph

1 4

1 6

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Outline

1 Introduction

2 Main result

3 Outline of proof

4 Application

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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.

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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.

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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