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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.

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Presentation on theme: "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."— Presentation transcript:

1 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

2 4 3 1 4 3 1 1 4 3 1 Input Graph G Acyclic Coloring of G 1 4 3 1 Acyclic Coloring 1 1 4 1 3 1 4 3 6/21/20112IWOCA 2011, Victoria 1 2 3 4 5 6 1 2 3 4 5 6

3 4 3 1 4 3 1 1 4 3 1 Input Graph G 1 4 3 1 1 1 4 1 3 1 4 3 Acyclic Coloring of a subdivision of G Why subdivision ? 6/21/20113IWOCA 2011, Victoria 1 2 3 4 5 6 1 2 3 4 5 6

4 4 3 1 4 3 1 1 4 3 1 Input Graph G Acyclic Coloring of a subdivision of G 1 4 3 1 Why subdivision ? 1 1 4 1 3 1 4 3 4 3 3 Division vertex 6/21/20114IWOCA 2011, Victoria 1 2 3 4 5 6 1 2 3 4 56 7

5 A subdivision G of K 5 Input graph K 5 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 graphs Division vertices correspond to the total number of bends in the polyline drawing. Straight-line drawing of G in 3D Poly-line drawing of K 5 in 3D 6/21/20115IWOCA 2011, Victoria

6 Previous Results Grunbaum1973Lower bound on acyclic colorings of planar graphs is 5 Borodin1979Every planar graph is acyclically 5-colorable Kostochka1978Deciding 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 6/21/20116IWOCA 2011, Victoria Ochem2005Testing acyclic 4-colorability is NP-complete for planar bipartite graphs with maximum degree 8

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/2 division vertices. Partial k-tree, k ≤ 8One 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 6/21/20117IWOCA 2011, Victoria

8 Some Observations 3 1 v u 1 v u 3 1 3 1 w w1w1 w2w2 w3w3 wnwn G G G / admits an acyclic 3-coloring G / G / 6/21/20118IWOCA 2011, Victoria

9 Some Observations 1 G G admits an acyclic 3-coloring with at most |E|-n subdivisions 1 2 3 2 1 3 3 2 2 1 2 2 1 Subdivision a b c d e f g h i j k l m n 2 l x 6/21/20119IWOCA 2011, Victoria G is a biconnected graph that has a non-trivial ear decomposition. Ear

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/2 division vertices. Partial k-tree, k ≤ 8One 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 6/21/201110IWOCA 2011, Victoria

11 14 15 8 Acyclic coloring of a 3-connected cubic graph 1 2 3 4 17 5 6 7 9 10 11 16 18 3 1 3 1 2 1 3 2 3 2 1 3 1 2 3 1 2 3 12 13 12 13 1 2 3 4 17 5 6 7 8 9 10 11 16 18 14 15 Subdivision Every 3-connected cubic graph admits an acyclic 3-coloring with at most |E| - n = 3n/2 – n = n/2 subdivisions 6/21/201111IWOCA 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/2 division vertices. Partial k-tree, k ≤ 8One 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 6/21/201112IWOCA 2011, Victoria

13 u Acyclic coloring of a partial k-tree, k ≤ 8 G 11 11 1 1 11 2 G / 6/21/201113IWOCA 2011, Victoria

14 u Acyclic coloring of a partial k-tree, k ≤ 8 G 11 2 1 2 1 1 2 3 G / 6/21/201114IWOCA 2011, Victoria

15 u Acyclic coloring of a partial k-tree, k ≤ 8 G 11 1 2 2 2 33 3 Every partial k-tree admits an acyclic 3-coloring for k ≤ 8 with at most |E| subdivisions G / 6/21/201115IWOCA 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/2 division vertices. Partial k-tree, k ≤ 8One 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 6/21/201116IWOCA 2011, Victoria

17 Acyclic 3-coloring of triangulated graphs 2 1 3 3 1 3 1 1 1 2 3 4 5 6 7 8 6/21/201117IWOCA 2011, Victoria

18 Acyclic 3-coloring of triangulated graphs 2 1 3 3 1 3 1 1 1 2 2 1 3 4 5 6 7 8 3 6/21/201118IWOCA 2011, Victoria

19 Acyclic 3-coloring of triangulated graphs 2 1 3 3 1 3 1 1 1 2 2 1 3 4 5 6 7 8 3 2 6/21/201119IWOCA 2011, Victoria

20 Acyclic 3-coloring of triangulated graphs 2 1 3 3 1 3 1 1 1 2 2 1 3 4 5 6 7 8 3 2 1 6/21/201120IWOCA 2011, Victoria

21 Acyclic 3-coloring of triangulated graphs 2 1 3 3 1 3 1 1 1 2 2 1 3 4 5 6 7 8 3 2 1 3 6/21/201121IWOCA 2011, Victoria

22 Acyclic 3-coloring of triangulated graphs 2 1 3 3 1 3 1 1 1 2 2 1 3 4 5 6 7 8 3 2 1 3 1 6/21/201122IWOCA 2011, Victoria

23 Acyclic 3-coloring of triangulated graphs 2 1 3 3 1 3 1 1 1 2 1 3 4 5 6 7 8 3 2 1 3 1 3 2 6/21/201123IWOCA 2011, Victoria

24 Acyclic 3-coloring of triangulated graphs 1 3 3 1 3 1 1 1 2 1 3 4 5 6 7 8 3 1 3 1 3 6/21/201124IWOCA 2011, Victoria

25 Acyclic 3-coloring of triangulated graphs 1 3 3 1 3 1 1 1 2 1 3 4 5 6 7 8 3 1 3 1 3 Internal Edge External Edge |E| division vertices 6/21/201125IWOCA 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/2 division vertices. Partial k-tree, k ≤ 8One 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 6/21/201126IWOCA 2011, Victoria

27 Acyclic 4-coloring of triangulated graphs 2 1 3 3 1 3 1 1 1 2 3 4 5 6 7 8 6/21/201127IWOCA 2011, Victoria

28 Acyclic 4-coloring of triangulated graphs 2 1 3 3 1 3 1 1 1 2 2 1 3 4 5 6 7 8 3 6/21/201128IWOCA 2011, Victoria

29 Acyclic 4-coloring of triangulated graphs 2 1 3 3 1 3 1 1 1 2 2 1 3 4 5 6 7 8 3 2 6/21/201129IWOCA 2011, Victoria

30 Acyclic 4-coloring of triangulated graphs 2 1 3 3 1 3 1 1 1 2 2 1 3 4 5 6 7 8 3 2 1 6/21/201130IWOCA 2011, Victoria

31 Acyclic 4-coloring of triangulated graphs 2 1 3 3 1 3 1 1 1 2 2 1 3 4 5 6 7 8 3 2 1 3 2 2 6/21/201131IWOCA 2011, Victoria

32 Acyclic 4-coloring of triangulated graphs 2 1 3 3 1 3 1 1 1 2 2 1 3 4 5 6 7 8 3 2 1 3 1 6/21/201132IWOCA 2011, Victoria

33 Acyclic 4-coloring of triangulated graphs 2 1 3 3 1 3 1 1 1 2 1 3 4 5 6 7 8 3 2 1 3 1 3 2 6/21/201133IWOCA 2011, Victoria

34 Acyclic 4-coloring of triangulated graphs 2 1 3 3 1 3 1 1 1 2 1 3 4 5 6 7 8 3 2 1 3 1 3 2 Number of division vertices is |E| - n 6/21/201134IWOCA 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/2 division vertices. Partial k-tree, k ≤ 8One 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 6/21/201135IWOCA 2011, Victoria

36 3 1 2 1 3 2 3 1 … 12313212313211 … 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 6/21/201136IWOCA 2011, Victoria [Angelini & Frati, 2010] Acyclic three coloring of a planar graph with degree at most 4 is NP-complete

37 3 1 2 A graph G with maximum degree four 2 1 3 2 3 1 3 1 2 2 How to color? Maximum degree of G / is 7 An acyclic four coloring of G / must ensure acyclic three coloring in G. G/G/ 1 Acyclic 4-coloring is NP-complete for graphs with maximum degree 7 6/21/201137IWOCA 2011, Victoria Acyclic three coloring of a graph with degree at most 4 is NP-complete

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/2 division vertices. Partial k-tree, k ≤ 8One 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 6/21/201138IWOCA 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}? 6/21/201139IWOCA 2011, Victoria

40 6/21/201140IWOCA 2011, Victoria

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