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Acyclic Colorings of Graph Subdivisions

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.

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Acyclic Colorings of Graph Subdivisions

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  1. Acyclic Colorings of GraphSubdivisions 1Debajyoti Mondal2Rahnuma Islam Nishat 2Sue Whitesides3Md. SaidurRahman 1University of Manitoba, Canada 2University of Victoria, Canada 3Bangladesh University of Engineering and Technology (BUET), Bangladesh

  2. AcyclicColoring 6 6 1 1 5 5 1 1 1 1 4 4 4 4 2 2 4 4 3 3 3 3 3 3 1 Input Graph G Acyclic Coloring of G 1 1 4 4 3 3 1 1 1 IWOCA 2011, Victoria

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

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

  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. Input graph K5 Straight-line drawing of G in 3D Poly-line drawing of K5 in 3D A subdivision G of K5 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 Ochem Kostochka 1978 2005 Deciding whether a graph admits an acyclic 3-coloring is NP-hard Testing acyclic 4-colorability is NP-complete for planar bipartite graphs with maximum degree 8 Angelini& Frati 2010 Every planar graph has a subdivision with one vertex per edge which is acyclically 3-colorable IWOCA 2011, Victoria

  7. Our Results Triangulated plane graph with n vertices 3-connected plane cubic graph with n vertices Partial k-tree, k ≤ 8 Triangulated plane graph with n vertices One subdivision per edge, Acyclically 4-colorable One subdivision per edge, Acyclically 3-colorable One subdivision per edge, Acyclically 3-colorable Acyclically 3-colorable, simpler proof, originally proved by Angelini & Frati, 2010 At most 2n − 6 division vertices. At most n/2 division vertices. Each edge has exactly one division vertex Each edge has exactly one division vertex Acyclic 4-coloring is NP-complete for graphs with maximum degree 7. IWOCA 2011, Victoria

  8. Some Observations w3 3 w2 wn 1 w 1 3 w1 3 v v u u G G 1 1 / / G G G/admits an acyclic 3-coloring IWOCA 2011, Victoria

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

  10. Our Results Triangulated plane graph with n vertices 3-connected plane cubic graph with n vertices Partial k-tree, k ≤ 8 Triangulated plane graph with n vertices One subdivision per edge, Acyclically 4-colorable One subdivision per edge, Acyclically 3-colorable One subdivision per edge, Acyclically 3-colorable Acyclically 3-colorable, simpler proof, originally proved by Angelini & Frati, 2010 At most 2n − 6 division vertices. At most n/2 division vertices. Each edge has exactly one division vertex Each edge has exactly one division vertex Acyclic 4-coloring is NP-complete for graphs with maximum degree 7. IWOCA 2011, Victoria

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

  12. Our Results Triangulated plane graph with n vertices 3-connected plane cubic graph with n vertices Partial k-tree, k ≤ 8 Triangulated plane graph with n vertices One subdivision per edge, Acyclically 4-colorable One subdivision per edge, Acyclically 3-colorable One subdivision per edge, Acyclically 3-colorable Acyclically 3-colorable, simpler proof, originally proved by Angelini & Frati, 2010 At most 2n − 6 division vertices. At most n/2 division vertices. Each edge has exactly one division vertex Each edge has exactly one division vertex Acyclic 4-coloring is NP-complete for graphs with maximum degree 7. IWOCA 2011, Victoria

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

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

  15. Acyclic coloring of a partial k-tree, k ≤ 8 u 3 3 3 1 1 2 1 2 2 G / G Every partial k-tree admits an acyclic 3-coloring for k ≤ 8 with at most |E| subdivisions IWOCA 2011, Victoria

  16. Our Results Triangulated plane graph with n vertices 3-connected plane cubic graph with n vertices Partial k-tree, k ≤ 8 Triangulated plane graph with n vertices One subdivision per edge, Acyclically 4-colorable One subdivision per edge, Acyclically 3-colorable One subdivision per edge, Acyclically 3-colorable Acyclically 3-colorable, simpler proof, originally proved by Angelini & Frati, 2010 At most 2n − 6 division vertices. At most n/2 division vertices. Each edge has exactly one division vertex Each edge has exactly one division vertex Acyclic 4-coloring is NP-complete for graphs with maximum degree 7. IWOCA 2011, Victoria

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

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

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

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

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

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

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

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

  25. Acyclic 3-coloring of triangulated graphs 3 3 8 Internal Edge External Edge |E| division vertices 6 7 1 1 3 1 5 1 1 4 3 3 3 3 1 2 1 1 IWOCA 2011, Victoria

  26. Our Results Triangulated plane graph with n vertices 3-connected plane cubic graph with n vertices Partial k-tree, k ≤ 8 Triangulated plane graph with n vertices One subdivision per edge, Acyclically 4-colorable One subdivision per edge, Acyclically 3-colorable One subdivision per edge, Acyclically 3-colorable Acyclically 3-colorable, simpler proof, originally proved by Angelini & Frati, 2010 At most 2n − 6 division vertices. At most n/2 division vertices. Each edge has exactly one division vertex Each edge has exactly one division vertex Acyclic 4-coloring is NP-complete for graphs with maximum degree 7. IWOCA 2011, Victoria

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

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

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

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

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

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

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

  34. Acyclic 4-coloring of triangulated graphs 3 3 8 Number of division vertices is |E| - n 6 7 1 1 3 1 5 1 1 4 2 3 3 3 3 1 2 1 1 2 2 IWOCA 2011, Victoria

  35. Our Results Triangulated plane graph with n vertices 3-connected plane cubic graph with n vertices Partial k-tree, k ≤ 8 Triangulated plane graph with n vertices One subdivision per edge, Acyclically 4-colorable One subdivision per edge, Acyclically 3-colorable One subdivision per edge, Acyclically 3-colorable Acyclically 3-colorable, simpler proof, originally proved by Angelini & Frati, 2010 At most 2n − 6 division vertices. At most n/2 division vertices. Each edge has exactly one division vertex Each edge has exactly one division vertex Acyclic 4-coloring is NP-complete for graphs with maximum degree 7. IWOCA 2011, Victoria

  36. Acyclic 4-coloring is NP-complete for graphs with maximum degree 7 [Angelini & Frati, 2010] Acyclic three coloring of a planar graph with degree at most 4 is NP-complete 3 Each of the blue vertices are of degree is 6 1 3 Infinite number of nodes with the same color at regular intervals … 1 2 3 2 1 1 1 3 2 3 1 2 … 1 1 3 2 3 1 2 IWOCA 2011, Victoria

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

  38. Summary of Our Results Triangulated plane graph with n vertices 3-connected plane cubic graph with n vertices Partial k-tree, k ≤ 8 Triangulated plane graph with n vertices One subdivision per edge, Acyclically 4-colorable One subdivision per edge, Acyclically 3-colorable One subdivision per edge, Acyclically 3-colorable Acyclically 3-colorable, simpler proof, originally proved by Angelini & Frati, 2010 At most 2n − 6 division vertices. At most n/2 division vertices. Each edge has exactly one division vertex Each edge has exactly one division vertex Acyclic 4-coloring is NP-complete for graphs with maximum degree 7. 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 cndivision vertices that is acyclicallyk-colorable, k ∈ {3,4}? IWOCA 2011, Victoria

  40. THANK YOU IWOCA 2011, Victoria

  41. 3 1 3 … 1 2 3 2 1 IWOCA 2011, Victoria

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