Acyclic coloring with few division vertices

Authors:
Debajyoti Mondal;Rahnuma Islam Nishat;Md. Saidur Rahman;Sue Whitesides
Affiliations:
Department of Computer Science, University of Manitoba, Canada;Department of Computer Science, University of Victoria, Canada;Graph Drawing and Information Visualization Laboratory, Department of Computer Science and Engineering, Bangladesh University of Engineering and Technology, Bangladesh;Department of Computer Science, University of Victoria, Canada
Venue:
Journal of Discrete Algorithms
Year:
2013

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Abstract

An acyclic k-coloring of a graph G is a mapping @f 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 @f does not contain any bichromatic cycle. In this paper we prove that every planar 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, n-3). On the other hand, we prove a 1.28n (respectively, 0.3n) lower bound on the number of division vertices for acyclic 3-colorings (respectively, 4-colorings) of planar graphs. Furthermore, we establish the NP-completeness of deciding acyclic 4-colorability for graphs with maximum degree 5 and for planar graphs with maximum degree 7.