# Acyclic Colorings of Graph Subdivisions

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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

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

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