Acyclic coloring with few division vertices

Debajyoti Mondal, Rahnuma Islam Nishat, Md. Saidur Rahman, Sue Whitesides
2013 Journal of Discrete Algorithms  
An acyclic k-coloring of a graph G is a mapping φ from the set of vertices of G to a set of k distinct colors such that no two adjacent vertices receive the same color and φ does not contain any bichromatic cycle. In this paper we prove that every triangulated plane graph with n vertices has a 1-subdivision that is acyclically 3-colorable (respectively, 4-colorable), where the number of division vertices is at most 2n − 5 (respectively, 1.5n − 3.5). Our results imply O(n log 16 n) and O(n log
more » ... n) upper bounds on volume of 3D polyline drawings of planar graphs, where each edge has at most one bend and the total number of bends are 2n − 5 and 1.5n − 3.5, respectively. On the other hand, we prove an 1.28n (respectively, 0.3n) lower bound on the number of division vertices for acyclic 3-colorings (respectively, 4-colorings) of triangulated planar graphs. Furthermore, we establish the NP-completeness of deciding acyclic 4-colorability for graphs with the maximum degree 5 and for planar graphs with the maximum degree 7.
doi:10.1016/j.jda.2013.08.002 fatcat:t465grxbofhl7hcik22p7kefry