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2-dipath and oriented L(2, 1)-labelings of somefamilies of oriented planar graphs

Sagnik Sen

Supervisors:

Prof. ric Sopena, Prof. Andr Raspaud, Prof. Arnaud Pcher

LaBRI, Universit de Bordeaux 1

31/08/2011

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

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Basic definitions

Definition

A k-L(2, 1)-labeling of a graph G is a function from the vertexset V(G) to the set{0, 1, ...., k} such that- | (u) (v) | 2 if u and v are at distance 1 in G,- | (u) (v) | 1 if u and v are at distance 2 in G.

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Basic definitions

Definition

A k-L(2, 1)-labeling of a graph G is a function from the vertexset V(G) to the set{0, 1, ...., k} such that- | (u) (v) | 2 if u and v are at distance 1 in G,- | (u) (v) | 1 if u and v are at distance 2 in G.

Span 2,1(G) = min{k | G has a k-L(2, 1)-labeling}.

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Basic definitions

Definition

A k-L(2, 1)-labeling of a graph G is a function from the vertexset V(G) to the set{0, 1, ...., k} such that- | (u) (v) | 2 if u and v are at distance 1 in G,- | (u) (v) | 1 if u and v are at distance 2 in G.

Span 2,1(G) = min{k | G has a k-L(2, 1)-labeling}.

3

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0

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Figure: Star

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Basic definitions

Definition

A 2-dipath k-L(2, 1)-labeling of an oriented graph G is a function from the vertex set V(G) to the set{0, 1, ...., k} such that- | (u) (v) | 2 if u and v are adjacent in G,- | (u) (v) | 1 if u and v are connected by a directed2-path in G.

2-dipath span2,1(G) = min{k | G has a 2-dipathk -L(2, 1)-labeling}.

2

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0

2

3

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3

Figure: StarSagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

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Basic definitions

Definition

A homomorphism f of an oriented graph G to an oriented graphH is a mapping f : V(G) V(H) such that xy A(G) impliesf(x)f(y) V(H).

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

B i d fi i i

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Basic definitions

Definition

A homomorphism f of an oriented graph G to an oriented graphH is a mapping f : V(G) V(H) such that xy A(G) impliesf(x)f(y) V(H).

302

2

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0

2

3

3

3

Figure: Homomorphism

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B i d fi iti

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Basic definitions

Definition

An oriented k-L(2, 1)-labeling of an oriented graph G is afunction from the vertex set V(G) to the set{0, 1, ...., k} suchthat

- is a 2-dipath k-L(2, 1)-labeling of G,- If xy and uv are two arcs in G then, (x) = (v) implies(y) = (u).

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

B i d fi iti

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Basic definitions

Definition

An oriented k-L(2, 1)-labeling of an oriented graph G is afunction from the vertex set V(G) to the set{0, 1, ...., k} suchthat

- is a 2-dipath k-L(2, 1)-labeling of G,- If xy and uv are two arcs in G then, (x) = (v) implies(y) = (u).

Oriented spano

2,

1(G) = min{k | G has an oriented

k -L(2, 1)-labeling}.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

B i lt

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Basic results

Lemma

2,1(G) o2,1(

G)

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

Basic res lts

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Basic results

Lemma

2,1(G) o2,1(

G)

Lemma

If there is a homomorphism f : G H theno2,1(G) o2,1(

H).

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

The existing results (Calamoneri and Sinaimeri (2010))

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The existing results (Calamoneri and Sinaimeri (2010))

Conjecture

2,

1(P5) = 5.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

The existing results (Calamoneri and Sinaimeri (2010))

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The existing results (Calamoneri and Sinaimeri (2010))

Conjecture

2,

1(P5) =5.

Theorem2,1(P11) 12.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

The existing results (Calamoneri and Sinaimeri (2010))

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The existing results (Calamoneri and Sinaimeri (2010))

Conjecture

2,

1(P5) =5.

Theorem2,1(P11) 12.

Theorem

2,1(P16) 8.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

The existing results (Calamoneri and Sinaimeri (2010))

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The existing results (Calamoneri and Sinaimeri (2010))

Conjecture

2,

1(P5

) = 5.

Theorem2,1(P11) 12.

Theorem

2,1(P16) 8.

Theorem

6 2,1(C) 8.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

Our improvements

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Our improvements

Theorem

6 2,1(P5).

Theorem

2,1(P11) 12.

Theorem

2,1(P16) 8.

Theorem

6 2,1(C) 8.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

Our improvements

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Our improvements

Theorem

6 2,1(P5).

Theorem

2,1(P11) o2,1(P11) 10.

Theorem2,1(P16) 8.

Theorem

6 2,1(C) 8.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

Our improvements

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Our improvements

Theorem

6 2,1(P5).

Theorem

2,1(P11) o2,1(P11) 10.

Theorem

2,1(P16) o2,1(P16) 7.

Theorem

6 2,1(C) 8.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

Our improvements

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Our improvements

Theorem

6 2,1(P5).

Theorem2,1(P11)

o2,1(P11) 10.

Theorem2,1(P16)

o2,1(P16) 7.

Theorem

2,1(C) = op,q(C) = 7.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

The existing results (Calamoneri and Sinaimeri (2010))

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The existing results (Calamoneri and Sinaimeri (2010))

Theorem

8 2,1(H) 10.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

The existing results (Calamoneri and Sinaimeri (2010))

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The existing results (Calamoneri and Sinaimeri (2010))

Theorem

8 2,1(H) 10.

Theorem

8 2,1(W) 9.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

Halin graph

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Halin graph

Theorem8 2,1(H) 10.

Theorem

2,1(W) = 8.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

Halin graph

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Halin graph

Theorem8 2,1(H) 10.

Theorem

2,1(W) = 8.

Theorem

7 2,1(Hli) 8.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

A 4-Nice graph

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A 4 Nice graph

4 3

7 0

5 2

6 1

Figure: B is a 4-nice graph.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

Cactus (lower bound proof)

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Cactus (lower bound proof)

x=2

Figure: The cactus H.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente

Thank you

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a you

Thank you for your attention.

Sagnik Sen 2-dipath and oriented L(2, 1)-labelings of some families of oriente