Three Colors Suffice: Conflict-Free Coloring of Planar Graphs
Name
Three colors suffice.pdf
Size
519.06 KB
Format
Adobe PDF
Checksum (MD5)
586250c575afab3947f50d5c8f5acbec
Author(s)
Abel, Zachary
Alvarez, Victor
Demaine, Erik D.
Fekete, Sándor P.
Gour, Aman
Hesterberg, Adam
Keldenich, Phillip
Scheffer, Christian
Date Issued
January 2017
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
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1137/1.9781611974782.127
