Conflict-Free Coloring of Graphs
Name
17m1146579.pdf
Description
Published version
Size
753.75 KB
Format
Adobe PDF
Checksum (MD5)
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Author(s)
Abel, Zachary R
Alvarez, Victor
Demaine, Erik D
Fekete, Sándor P.
Gour, Aman
Hesterberg, Adam Classen
Keldenich, Phillip
Scheffer, Christian
Date Issued
November 27, 2018
Journal
SIAM Journal on Discrete Mathematics
Publisher
Society for Industrial & Applied Mathematics
Citation
Abel, Zachary et al. "Conflict-Free Coloring of Graphs." SIAM Journal on Discrete Mathematics 32, 4 (2018): 2675–2702 © 2018 The Author(s)
Version
Final published version
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 χCF(G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N[v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N(v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K [subscript k+1] as a minor, then χCF(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 outerplanar 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. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k ∈ {1,2,3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general planar graph has a conflict-free coloring with at most eight colors. Keywords: conflict-free coloring; planar graphs; complexity; worst-case bound
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1137/17m1146579
