## Three Colors Suffice: Conflict-Free Coloring of Planar Graphs

##### Author(s)

Abel, Zachary; Alvarez, Victor; Demaine, Erik D.; Fekete, Sándor P.; Gour, Aman; Hesterberg, Adam; Keldenich, Phillip; Scheffer, Christian; ... Show more Show less
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A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number x[subscript CF](G) (the smallest k for which conflict-free k-colorings exist), with a focus on planar graphs.
For general graphs, we prove the conflict-free variant of the famous Hadwiger Conjecture: If G does not contain K[subscript k+1] as a minor, then x[subscript CF](G) < k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. In addition, we give a complete characterization of the algorithmic/computational complexity of conflict-free coloring. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color, while for outer- planar graphs, this can be decided in polynomial time. Furthermore, it is NP-complete to decide whether a planar graph has a conflict-free coloring with two colors, while for outerplanar graphs, two colors always suffice. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs.

##### Date issued

2017-01##### Department

Massachusetts Institute of Technology. Department of Mathematics; Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory##### Journal

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

##### Citation

Abel, Zachary, et al. "Three Colors Suffice: Conflict-Free Coloring of Planar Graphs." Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, 16-19 January, 2017, Barcelona, Spain, Society for Industrial and Applied Mathematics, 2017, pp. 1951–63.

Version: Author's final manuscript

##### ISBN

978-1-61197-478-2