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

Input Graph G Acyclic Coloring of G Acyclic Coloring /21/20112IWOCA 2011, Victoria

Input Graph G Acyclic Coloring of a subdivision of G Why subdivision ? 6/21/20113IWOCA 2011, Victoria

Input Graph G Acyclic Coloring of a subdivision of G Why subdivision ? Division vertex 6/21/20114IWOCA 2011, Victoria

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

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

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

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

Some Observations 1 G G admits an acyclic 3-coloring with at most |E|-n subdivisions 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

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

Acyclic coloring of a 3-connected cubic graph 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

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

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

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

u Acyclic coloring of a partial k-tree, k ≤ 8 G Every partial k-tree admits an acyclic 3-coloring for k ≤ 8 with at most |E| subdivisions G / 6/21/201115IWOCA 2011, Victoria

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

Acyclic 3-coloring of triangulated graphs /21/201117IWOCA 2011, Victoria

Acyclic 3-coloring of triangulated graphs /21/201118IWOCA 2011, Victoria

Acyclic 3-coloring of triangulated graphs /21/201119IWOCA 2011, Victoria

Acyclic 3-coloring of triangulated graphs /21/201120IWOCA 2011, Victoria

Acyclic 3-coloring of triangulated graphs /21/201121IWOCA 2011, Victoria

Acyclic 3-coloring of triangulated graphs /21/201122IWOCA 2011, Victoria

Acyclic 3-coloring of triangulated graphs /21/201123IWOCA 2011, Victoria

Acyclic 3-coloring of triangulated graphs /21/201124IWOCA 2011, Victoria

Acyclic 3-coloring of triangulated graphs Internal Edge External Edge |E| division vertices 6/21/201125IWOCA 2011, Victoria

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

Acyclic 4-coloring of triangulated graphs /21/201127IWOCA 2011, Victoria

Acyclic 4-coloring of triangulated graphs /21/201128IWOCA 2011, Victoria

Acyclic 4-coloring of triangulated graphs /21/201129IWOCA 2011, Victoria

Acyclic 4-coloring of triangulated graphs /21/201130IWOCA 2011, Victoria

Acyclic 4-coloring of triangulated graphs /21/201131IWOCA 2011, Victoria

Acyclic 4-coloring of triangulated graphs /21/201132IWOCA 2011, Victoria

Acyclic 4-coloring of triangulated graphs /21/201133IWOCA 2011, Victoria

Acyclic 4-coloring of triangulated graphs Number of division vertices is |E| - n 6/21/201134IWOCA 2011, Victoria

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

… … 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

3 1 2 A graph G with maximum degree four 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

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

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

6/21/201140IWOCA 2011, Victoria