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|a Abel, Zachary R
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|a Massachusetts Institute of Technology. Department of Mathematics
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|a Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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|a Alvarez, Victor
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|a Demaine, Erik D
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|a Fekete, Sándor P.
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|a Gour, Aman
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|a Hesterberg, Adam Classen
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|a Keldenich, Phillip
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|a Scheffer, Christian
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|a Conflict-Free Coloring of Graphs
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|b Society for Industrial & Applied Mathematics,
|c 2019-11-15T16:48:34Z.
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|z Get fulltext
|u https://hdl.handle.net/1721.1/122951
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|a 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
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|a en
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|a Article
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|t SIAM Journal on Discrete Mathematics
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