Acyclic Colorings of Graph Subdivisions
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Transcript of Acyclic Colorings of Graph Subdivisions
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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
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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
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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
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Ochem 2005 Testing acyclic 4-colorability is NP-complete for planar bipartite graphs with maximum degree 8
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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
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Some Observations
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G/ admits an acyclic 3-coloring
G /G /
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Some Observations
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G admits an acyclic 3-coloring with at most |E|-n subdivisions
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G is a biconnected graph that has a non-trivial ear decomposition.
Ear
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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
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Acyclic coloring of a 3-connected cubic graph
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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
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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
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Acyclic coloring of a partial k-tree, k ≤ 8
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Acyclic coloring of a partial k-tree, k ≤ 8
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Acyclic coloring of a partial k-tree, k ≤ 8
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Every partial k-tree admits an acyclic 3-coloring for k ≤ 8 with at most |E| subdivisions
G /
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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 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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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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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
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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
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Acyclic 4-coloring of triangulated graphs
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Acyclic 4-coloring of triangulated graphs
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Acyclic 4-coloring of triangulated graphs
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Acyclic 4-coloring of triangulated graphs
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Acyclic 4-coloring of triangulated graphs
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Acyclic 4-coloring of triangulated graphs
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Acyclic 4-coloring of triangulated graphs
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Acyclic 4-coloring of triangulated graphs
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Number of division vertices is |E| - n
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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
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Infinite number of nodes with the same color at regular
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Each of the blue vertices are of degree is 6
Acyclic 4-coloring is NP-complete for graphs with maximum degree 7
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[Angelini & Frati, 2010] Acyclic three coloring of a planar graph with degree at most 4 is NP-complete
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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.
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Acyclic three coloring of a graph with degree at most
4 is NP-complete
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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
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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}?
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THANK YOU
6/21/2011
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