1 debajyoti mondal 2 rahnuma islam nishat 2 sue whitesides 3 md. saidur rahman 1 university of...
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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
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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Input Graph G
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Acyclic Coloring ofa subdivision of G
Why subdivision ?
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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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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
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
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
IWOCA 2011, Victoria 8
Some Observations
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u 1
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1w
w1
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wn
G G
G/ admits an acyclic 3-coloring
G /G /
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IWOCA 2011, Victoria 9
Some Observations
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G
G admits an acyclic 3-coloring with at most |E|-n subdivisions
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Subdivisiona
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cd
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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
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
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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
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
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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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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u
Acyclic coloring of a partial k-tree, k ≤ 8
G
1 1 1 2 2 23 3
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Every partial k-tree admits an acyclic 3-coloring for k ≤ 8 with at most |E| subdivisions
G /
6/21/2011 15IWOCA 2011, Victoria
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
IWOCA 2011, Victoria 17
Acyclic 3-coloring of triangulated graphs
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IWOCA 2011, Victoria 18
Acyclic 3-coloring of triangulated graphs
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IWOCA 2011, Victoria 19
Acyclic 3-coloring of triangulated graphs
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Acyclic 3-coloring of triangulated graphs
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Acyclic 3-coloring of triangulated graphs
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IWOCA 2011, Victoria 22
Acyclic 3-coloring of triangulated graphs
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IWOCA 2011, Victoria 23
Acyclic 3-coloring of triangulated graphs
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Acyclic 3-coloring of triangulated graphs
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Acyclic 3-coloring of triangulated graphs
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Internal Edge
External Edge
|E| division vertices
6/21/2011
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
IWOCA 2011, Victoria 27
Acyclic 4-coloring of triangulated graphs
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4 5
6 7
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6/21/2011
IWOCA 2011, Victoria 28
Acyclic 4-coloring of triangulated graphs
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1 221
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IWOCA 2011, Victoria 29
Acyclic 4-coloring of triangulated graphs
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IWOCA 2011, Victoria 30
Acyclic 4-coloring of triangulated graphs
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IWOCA 2011, Victoria 31
Acyclic 4-coloring of triangulated graphs
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IWOCA 2011, Victoria 32
Acyclic 4-coloring of triangulated graphs
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IWOCA 2011, Victoria 33
Acyclic 4-coloring of triangulated graphs
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IWOCA 2011, Victoria 34
Acyclic 4-coloring of triangulated graphs
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Number of division vertices is |E| - n
6/21/2011
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
IWOCA 2011, Victoria 36
3
1 2
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…
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
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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/
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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
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
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