acyclic colorings of graph subdivisions

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Acyclic Colorings of Graph Subdivisions 1 Debajyoti Mondal 2 Rahnuma Islam Nishat 2 Sue Whitesides 3 Md. Saidur Rahman 1 University of Manitoba, Canada 2 University of Victoria, Canada 3 Bangladesh University of Engineering and Technology (BUET), Bangladesh

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Acyclic Colorings of Graph Subdivisions. 1 Debajyoti Mondal 2 Rahnuma Islam Nishat 2 Sue Whitesides 3 Md . Saidur Rahman. 1 University of Manitoba, Canada 2 University of Victoria, Canada 3 Bangladesh University of Engineering and Technology (BUET), Bangladesh. Acyclic Coloring. 6. - PowerPoint PPT Presentation

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Page 1: Acyclic  Colorings  of  Graph Subdivisions

Acyclic Colorings of Graph Subdivisions

1Debajyoti Mondal 2Rahnuma Islam Nishat2Sue Whitesides 3Md. Saidur Rahman

1University of Manitoba, Canada2University of Victoria, Canada

3Bangladesh University of Engineering and Technology (BUET), Bangladesh

Page 2: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 2

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Input Graph G Acyclic Coloring of G

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Acyclic Coloring

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Page 3: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 3

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Input Graph G

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Acyclic Coloring ofa subdivision of G

Why subdivision ?

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Page 4: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 4

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Input Graph G Acyclic Coloring ofa subdivision of G

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Why subdivision ?

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Division vertex

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Page 5: Acyclic  Colorings  of  Graph Subdivisions

A subdivision G of K5

Input graph K5

Why subdivision ?

Acyclic coloring of planar graphs

Upper bounds on the volume of 3-dimensional straight-line grid drawings of planar graphs

Acyclic coloring of planar graph subdivisions Upper bounds on the volume of 3-dimensional polyline grid drawings of planar graphsDivision vertices correspond to the total number of bends in the polyline drawing.

Straight-line drawing of G in 3D

Poly-line drawing of K5 in 3D

6/21/2011 5IWOCA 2011, Victoria

Page 6: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 6

Previous Results

Grunbaum 1973 Lower bound on acyclic colorings of planar graphs is 5

Borodin 1979 Every planar graph is acyclically 5-colorable

Kostochka 1978 Deciding whether a graph admits an acyclic 3-coloring is NP-hard

2010Angelini & Frati

Every planar graph has a subdivision with one vertex per edge which is acyclically 3-colorable

6/21/2011

Ochem 2005 Testing acyclic 4-colorability is NP-complete for planar bipartite graphs with maximum degree 8

Page 7: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 7

Triangulated plane graph with n

vertices

One subdivision per edge,

Acyclically 4-colorable

At most 2n − 6 division vertices.

Our Results

Acyclic 4-coloring is NP-complete for graphs with maximum degree 7.

3-connected plane cubic graph with n

vertices

One subdivision per edge,

Acyclically 3-colorable

At most n/2division vertices.

Partial k-tree, k ≤ 8 One subdivision per edge,

Acyclically 3-colorable

Each edge has exactly one

division vertex

Triangulated plane graph with n

vertices

Acyclically 3-colorable, simpler proof, originally

proved by Angelini & Frati, 2010

Each edge has exactly one

division vertex

6/21/2011

Page 8: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 8

Some Observations

3

1

v

u 1

v

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1w

w1

w2

w3

wn

G G

G/ admits an acyclic 3-coloring

G /G /

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Page 9: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 9

Some Observations

1

G

G admits an acyclic 3-coloring with at most |E|-n subdivisions

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Subdivisiona

b

cd

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f

g

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ij

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2 l

x

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G is a biconnected graph that has a non-trivial ear decomposition.

Ear

Page 10: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 10

Triangulated plane graph with n

vertices

One subdivision per edge,

Acyclically 4-colorable

At most 2n − 6 division vertices.

Our Results

Acyclic 4-coloring is NP-complete for graphs with maximum degree 7.

3-connected plane cubic graph with n

vertices

One subdivision per edge,

Acyclically 3-colorable

At most n/2division vertices.

Partial k-tree, k ≤ 8 One subdivision per edge,

Acyclically 3-colorable

Each edge has exactly one

division vertex

Triangulated plane graph with n

vertices

Acyclically 3-colorable, simpler proof, originally

proved by Angelini & Frati, 2010

Each edge has exactly one

division vertex

6/21/2011

Page 11: Acyclic  Colorings  of  Graph Subdivisions

1415

8

Acyclic coloring of a 3-connected cubic graph

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7910

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16183

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Subdivision

Subdivision

Every 3-connected cubic graph admits an acyclic 3-coloring with at most |E| - n = 3n/2 – n = n/2

subdivisions6/21/2011 11IWOCA 2011, Victoria

Page 12: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 12

Triangulated plane graph with n

vertices

One subdivision per edge,

Acyclically 4-colorable

At most 2n − 6 division vertices.

Our Results

Acyclic 4-coloring is NP-complete for graphs with maximum degree 7.

3-connected plane cubic graph with n

vertices

One subdivision per edge,

Acyclically 3-colorable

At most n/2division vertices.

Partial k-tree, k ≤ 8 One subdivision per edge,

Acyclically 3-colorable

Each edge has exactly one

division vertex

Triangulated plane graph with n

vertices

Acyclically 3-colorable, simpler proof, originally

proved by Angelini & Frati, 2010

Each edge has exactly one

division vertex

6/21/2011

Page 13: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 13

u

Acyclic coloring of a partial k-tree, k ≤ 8

G

1 1 1 1 1 11 1

2

G /

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Page 14: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 14

u

Acyclic coloring of a partial k-tree, k ≤ 8

G

1 1 2 1 2 11 2

3

G /

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Page 15: Acyclic  Colorings  of  Graph Subdivisions

u

Acyclic coloring of a partial k-tree, k ≤ 8

G

1 1 1 2 2 23 3

3

Every partial k-tree admits an acyclic 3-coloring for k ≤ 8 with at most |E| subdivisions

G /

6/21/2011 15IWOCA 2011, Victoria

Page 16: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 16

Triangulated plane graph with n

vertices

One subdivision per edge,

Acyclically 4-colorable

At most 2n − 6 division vertices.

Our Results

Acyclic 4-coloring is NP-complete for graphs with maximum degree 7.

3-connected plane cubic graph with n

vertices

One subdivision per edge,

Acyclically 3-colorable

At most n/2division vertices.

Partial k-tree, k ≤ 8 One subdivision per edge,

Acyclically 3-colorable

Each edge has exactly one

division vertex

Triangulated plane graph with n

vertices

Acyclically 3-colorable, simpler proof, originally

proved by Angelini & Frati, 2010

Each edge has exactly one

division vertex

6/21/2011

Page 17: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 17

Acyclic 3-coloring of triangulated graphs

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Page 18: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 18

Acyclic 3-coloring of triangulated graphs

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1 221

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Page 19: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 19

Acyclic 3-coloring of triangulated graphs

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Page 20: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 20

Acyclic 3-coloring of triangulated graphs

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Page 21: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 21

Acyclic 3-coloring of triangulated graphs

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Page 22: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 22

Acyclic 3-coloring of triangulated graphs

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Page 23: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 23

Acyclic 3-coloring of triangulated graphs

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Page 24: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 24

Acyclic 3-coloring of triangulated graphs

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Page 25: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 25

Acyclic 3-coloring of triangulated graphs

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Internal Edge

External Edge

|E| division vertices

6/21/2011

Page 26: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 26

Triangulated plane graph with n

vertices

One subdivision per edge,

Acyclically 4-colorable

At most 2n − 6 division vertices.

Our Results

Acyclic 4-coloring is NP-complete for graphs with maximum degree 7.

3-connected plane cubic graph with n

vertices

One subdivision per edge,

Acyclically 3-colorable

At most n/2division vertices.

Partial k-tree, k ≤ 8 One subdivision per edge,

Acyclically 3-colorable

Each edge has exactly one

division vertex

Triangulated plane graph with n

vertices

Acyclically 3-colorable, simpler proof, originally

proved by Angelini & Frati, 2010

Each edge has exactly one

division vertex

6/21/2011

Page 27: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 27

Acyclic 4-coloring of triangulated graphs

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6 7

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Page 28: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 28

Acyclic 4-coloring of triangulated graphs

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Page 29: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 29

Acyclic 4-coloring of triangulated graphs

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Page 30: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 30

Acyclic 4-coloring of triangulated graphs

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Page 31: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 31

Acyclic 4-coloring of triangulated graphs

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Page 32: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 32

Acyclic 4-coloring of triangulated graphs

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Page 33: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 33

Acyclic 4-coloring of triangulated graphs

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Page 34: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 34

Acyclic 4-coloring of triangulated graphs

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Number of division vertices is |E| - n

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Page 35: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 35

Triangulated plane graph with n

vertices

One subdivision per edge,

Acyclically 4-colorable

At most 2n − 6 division vertices.

Our Results

Acyclic 4-coloring is NP-complete for graphs with maximum degree 7.

3-connected plane cubic graph with n

vertices

One subdivision per edge,

Acyclically 3-colorable

At most n/2division vertices.

Partial k-tree, k ≤ 8 One subdivision per edge,

Acyclically 3-colorable

Each edge has exactly one

division vertex

Triangulated plane graph with n

vertices

Acyclically 3-colorable, simpler proof, originally

proved by Angelini & Frati, 2010

Each edge has exactly one

division vertex

6/21/2011

Page 36: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 36

3

1 2

1

3

2

3

1

1 231 3 21 231 3 21 1 …

Infinite number of nodes with the same color at regular

intervals

Each of the blue vertices are of degree is 6

Acyclic 4-coloring is NP-complete for graphs with maximum degree 7

6/21/2011

[Angelini & Frati, 2010] Acyclic three coloring of a planar graph with degree at most 4 is NP-complete

Page 37: Acyclic  Colorings  of  Graph Subdivisions

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1 2A graph G with

maximum degree four

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How to color?

Maximum degree of G/ is 7An acyclic four coloring of G/ must ensure acyclic three coloring in G.

G/

1

Acyclic 4-coloring is NP-complete for graphs with maximum degree 7

6/21/2011 37IWOCA 2011, Victoria

Acyclic three coloring of a graph with degree at most

4 is NP-complete

Page 38: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 38

Triangulated plane graph with n

vertices

One subdivision per edge,

Acyclically 4-colorable

At most 2n − 6 division vertices.

Summary of Our Results

Acyclic 4-coloring is NP-complete for graphs with maximum degree 7.

3-connected plane cubic graph with n

vertices

One subdivision per edge,

Acyclically 3-colorable

At most n/2division vertices.

Partial k-tree, k ≤ 8 One subdivision per edge,

Acyclically 3-colorable

Each edge has exactly one

division vertex

Triangulated plane graph with n

vertices

Acyclically 3-colorable, simpler proof, originally

proved by Angelini & Frati, 2010

Each edge has exactly one

division vertex

6/21/2011

Page 39: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 39

Open Problems

What is the complexity of acyclic 4-colorings for graphs with maximum

degree less than 7?

What is the minimum positive constant c, such that every triangulated plane graph with n vertices admits a subdivision with at most cn

division vertices that is acyclically k-colorable, k ∈ {3,4}?

6/21/2011

Page 40: Acyclic  Colorings  of  Graph Subdivisions

IWOCA 2011, Victoria 40

THANK YOU

6/21/2011