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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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Acyclic Colorings of Graph Subdivisions1Debajyoti Mondal 2Rahnuma Islam Nishat

2Sue Whitesides 3Md. Saidur Rahman1University of Manitoba, Canada2University of Victoria, Canada3Bangladesh University of Engineering and Technology (BUET), Bangladesh

4314311431Input Graph GAcyclic Coloring of G1431Acyclic Coloring114131436/21/20112IWOCA 2011, Victoria1234561234564314311431Input Graph G143111413143Acyclic Coloring ofa subdivision of GWhy subdivision ?6/21/20113IWOCA 2011, Victoria1234561234564314311431Input Graph GAcyclic Coloring ofa subdivision of G1431Why subdivision ?11413143433Division vertex6/21/20114IWOCA 2011, Victoria1234561234567A subdivision G of K5Input graph K5Why 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 3DPoly-line drawing of K5 in 3D6/21/20115IWOCA 2011, VictoriaPrevious ResultsGrunbaum1973Lower bound on acyclic colorings of planar graphs is 5Borodin1979Every planar graph is acyclically 5-colorableKostochka1978Deciding whether a graph admits an acyclic 3-coloring is NP-hard2010Angelini & FratiEvery planar graph has a subdivision with one vertex per edge which is acyclically 3-colorable6/21/20116IWOCA 2011, VictoriaOchem2005Testing acyclic 4-colorability is NP-complete for planar bipartite graphs with maximum degree 8Triangulated plane graph with n verticesOne subdivision per edge,Acyclically 4-colorableAt most 2n 6 division vertices.Our ResultsAcyclic 4-coloring is NP-complete for graphs with maximum degree 7.3-connected plane cubic graph with n verticesOne subdivision per edge,Acyclically 3-colorableAt most n/2division vertices.Partial k-tree, k 8One subdivision per edge,Acyclically 3-colorableEach edge has exactly one division vertexTriangulated plane graph with n verticesAcyclically 3-colorable, simpler proof, originally proved by Angelini & Frati, 2010 Each edge has exactly one division vertex6/21/20117IWOCA 2011, VictoriaSome Observations31vu1vu3131ww1w2w3wnGGG/ admits an acyclic 3-coloringG/G/6/21/20118IWOCA 2011, VictoriaSome Observations1GG admits an acyclic 3-coloring with at most |E|-n subdivisions1232133221221Subdivisionabcdefghijklmn2lx6/21/20119IWOCA 2011, VictoriaG is a biconnected graph that has a non-trivial ear decomposition.EarTriangulated plane graph with n verticesOne subdivision per edge,Acyclically 4-colorableAt most 2n 6 division vertices.Our ResultsAcyclic 4-coloring is NP-complete for graphs with maximum degree 7.3-connected plane cubic graph with n verticesOne subdivision per edge,Acyclically 3-colorableAt most n/2division vertices.Partial k-tree, k 8One subdivision per edge,Acyclically 3-colorableEach edge has exactly one division vertexTriangulated plane graph with n verticesAcyclically 3-colorable, simpler proof, originally proved by Angelini & Frati, 2010 Each edge has exactly one division vertex6/21/201110IWOCA 2011, Victoria14158Acyclic coloring of a 3-connected cubic graph1234175679101116183131213232131231231213121312341756789101116181415SubdivisionSubdivisionEvery 3-connected cubic graph admits an acyclic 3-coloring with at most |E| - n = 3n/2 n = n/2 subdivisions6/21/201111IWOCA 2011, VictoriaTriangulated plane graph with n verticesOne subdivision per edge,Acyclically 4-colorableAt most 2n 6 division vertices.Our ResultsAcyclic 4-coloring is NP-complete for graphs with maximum degree 7.3-connected plane cubic graph with n verticesOne subdivision per edge,Acyclically 3-colorableAt most n/2division vertices.Partial k-tree, k 8One subdivision per edge,Acyclically 3-colorableEach edge has exactly one division vertexTriangulated plane graph with n verticesAcyclically 3-colorable, simpler proof, originally proved by Angelini & Frati, 2010 Each edge has exactly one division vertex6/21/201112IWOCA 2011, VictoriauAcyclic coloring of a partial k-tree, k 8G111111112G/6/21/201113IWOCA 2011, VictoriauAcyclic coloring of a partial k-tree, k 8G112121123G/6/21/201114IWOCA 2011, VictoriauAcyclic coloring of a partial k-tree, k 8G111222333Every partial k-tree admits an acyclic 3-coloring for k 8 with at most |E| subdivisionsG/6/21/201115IWOCA 2011, VictoriaTriangulated plane graph with n verticesOne subdivision per edge,Acyclically 4-colorableAt most 2n 6 division vertices.Our ResultsAcyclic 4-coloring is NP-complete for graphs with maximum degree 7.3-connected plane cubic graph with n verticesOne subdivision per edge,Acyclically 3-colorableAt most n/2division vertices.Partial k-tree, k 8One subdivision per edge,Acyclically 3-colorableEach edge has exactly one division vertexTriangulated plane graph with n verticesAcyclically 3-colorable, simpler proof, originally proved by Angelini & Frati, 2010 Each edge has exactly one division vertex6/21/201116IWOCA 2011, VictoriaAcyclic 3-coloring of triangulated graphs21331311123456786/21/201117IWOCA 2011, VictoriaAcyclic 3-coloring of triangulated graphs21331311122134567836/21/201118IWOCA 2011, VictoriaAcyclic 3-coloring of triangulated graphs213313111221345678326/21/201119IWOCA 2011, VictoriaAcyclic 3-coloring of triangulated graphs2133131112213456783216/21/201120IWOCA 2011, VictoriaAcyclic 3-coloring of triangulated graphs21331311122134567832136/21/201121IWOCA 2011, VictoriaAcyclic 3-coloring of triangulated graphs213313111221345678321316/21/201122IWOCA 2011, VictoriaAcyclic 3-coloring of triangulated graphs2133131112134567832131326/21/201123IWOCA 2011, VictoriaAcyclic 3-coloring of triangulated graphs 13313111213456783 1313 6/21/201124IWOCA 2011, VictoriaAcyclic 3-coloring of triangulated graphs 13313111213456783 1313 Internal EdgeExternal Edge|E| division vertices6/21/201125IWOCA 2011, VictoriaTriangulated plane graph with n verticesOne subdivision per edge,Acyclically 4-colorableAt most 2n 6 division vertices.Our ResultsAcyclic 4-coloring is NP-complete for graphs with maximum degree 7.3-connected plane cubic graph with n verticesOne subdivision per edge,Acyclically 3-colorableAt most n/2division vertices.Partial k-tree, k 8One subdivision per edge,Acyclically 3-colorableEach edge has exactly one division vertexTriangulated plane graph with n verticesAcyclically 3-colorable, simpler proof, originally proved by Angelini & Frati, 2010 Each edge has exactly one division vertex6/21/201126IWOCA 2011, VictoriaAcyclic 4-coloring of triangulated graphs21331311123456786/21/201127IWOCA 2011, VictoriaAcyclic 4-coloring of triangulated graphs21331311122134567836/21/201128IWOCA 2011, VictoriaAcyclic 4-coloring of triangulated graphs213313111221345678326/21/201129IWOCA 2011, VictoriaAcyclic 4-coloring of triangulated graphs2133131112213456783216/21/201130IWOCA 2011, VictoriaAcyclic 4-coloring of triangulated graphs213313111221345678321326/21/201131IWOCA 2011, VictoriaAcyclic 4-coloring of triangulated graphs213313111221345678321316/21/201132IWOCA 2011, VictoriaAcyclic 4-coloring of triangulated graphs2133131112134567832131326/21/201133IWOCA 2011, VictoriaAcyclic 4-coloring of triangulated graphs213313111213456783213132Number of division vertices is |E| - n6/21/201134IWOCA 2011, VictoriaTriangulated plane graph with n verticesOne subdivision per edge,Acyclically 4-colorableAt most 2n 6 division vertices.Our ResultsAcyclic 4-coloring is NP-complete for graphs with maximum degree 7.3-connected plane cubic graph with n verticesOne subdivision per edge,Acyclically 3-colorableAt most n/2division vertices.Partial k-tree, k 8One subdivision per edge,Acyclically 3-colorableEach edge has exactly one division vertexTriangulated plane graph with n verticesAcyclically 3-colorable, simpler proof, originally proved by Angelini & Frati, 2010 Each edge has exactly one division vertex6/21/201135IWOCA 2011, Victoria3121323112313212313211 Infinite number of nodes with the same color at regular intervals Each of the blue vertices are of degree is 6Acyclic 4-coloring is NP-complete for graphs with maximum degree 76/21/201136IWOCA 2011, Victoria[Angelini & Frati, 2010] Acyclic three coloring of a planar graph with degree at most 4 is NP-complete312A graph G with maximum degree four2132313122How to color?Maximum degree of G/ is 7An acyclic four coloring of G/ must ensure acyclic three coloring in G.G/1Acyclic 4-coloring is NP-complete for graphs with maximum degree 76/21/201137IWOCA 2011, VictoriaAcyclic three coloring of a graph with degree at most 4 is NP-completeTriangulated plane graph with n verticesOne subdivision per edge,Acyclically 4-colorableAt most 2n 6 division vertices.Summary of Our ResultsAcyclic 4-coloring is NP-complete for graphs with maximum degree 7.3-connected plane cubic graph with n verticesOne subdivision per edge,Acyclically 3-colorableAt most n/2division vertices.Partial k-tree, k 8One subdivision per edge,Acyclically 3-colorableEach edge has exactly one division vertexTriangulated plane graph with n verticesAcyclically 3-colorable, simpler proof, originally proved by Angelini & Frati, 2010 Each edge has exactly one division vertex6/21/201138IWOCA 2011, VictoriaOpen ProblemsWhat 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, VictoriaTHANK YOU6/21/201140IWOCA 2011, Victoria6/21/2011IWOCA 2011, Victoria4131213231