Conflict-Free Coloring Made Stronger

Aigner-Horev, Elad; Krakovski, Roi; Smorodinsky, Shakhar

doi:10.1007/978-3-642-13731-0_11

“…Furthermore, for unit disks, Lev-Tov and Peleg [24] present an O(1)-approximation algorithm for the number of colors. Horev et al [22] extend this by showing that any set of n disks can be colored with O(k log n) colors, even if every point must see k distinct unique colors. Abam et al [1] discuss the problem in the context of cellular networks where the network has to be reliable even if some number of base stations fault, giving worst-case bounds for the number of colors required.…”

Section: Related Workmentioning

confidence: 95%

Conflict-Free Coloring of Graphs

Abel¹,

Álvarez²,

Demaine³

et al. 2018

SIAM J. Discrete Math.

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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 χ CF (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 k+1 as a minor, then χ 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 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.2. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color.For outerplanar graphs, this question can be decided in polynomial time.

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“…Furthermore, for unit disks, Lev-Tov and Peleg [23] present an O(1)-approximation algorithm for the number of colors. Horev et al [21] extend this by showing that any set of n disks can be colored with O(k log n) colors, even if every point must see k distinct unique colors. Abam et al [1] discuss the problem in the context of cellular networks where the network has to be reliable even if some number of base stations fault, giving worst-case bounds for the number of colors required.…”

Section: Related Workmentioning

confidence: 95%

Three Colors Suffice: Conflict-Free Coloring of Planar Graphs

Abel¹,

Álvarez²,

Gour³

et al. 2017

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

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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 χ 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 k+1 as a minor, then χ 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 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.

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“…We remark that Horev et al [16] show that χ kscf (H n ) ≤ k log n (as a special case of a more general framework), and Gargano and Rescigno [14] show that…”

Section: S(q))mentioning

confidence: 95%

On variants of conflict-free-coloring for hypergraphs

Cui

¹

,

Hu

²

2017

Discrete Applied Mathematics

1

0

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Conflict-free coloring is a kind of vertex coloring of hypergraphs requiring each hyperedge to have a color which appears only on one vertex. More generally, for a positive integer k there are k-conflict-free colorings (k-CF-colorings for short) and k-strong-conflict-free colorings (k-SCF-colorings for short). Let H n be the hypergraph of which the vertex-set is V n = {1, 2, . . . , n} and the hyperedge-set E n is the set of all (non-empty) subsets of V n consisting of consecutive elements of V n . Firstly, we study the k-SCF-coloring of H n , give the exact k-SCF-coloring chromatic number of H n for k = 2, 3, and present upper and lower bounds of the k-SCF-coloring chromatic number of H n for all k. Secondly, we give the exact k-CF-coloring chromatic number of H n for all k.

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Conflict-Free Coloring Made Stronger

Cited by 20 publications

References 24 publications

Conflict-Free Coloring of Graphs

Conflict-Free Coloring of Graphs

Three Colors Suffice: Conflict-Free Coloring of Planar Graphs

On variants of conflict-free-coloring for hypergraphs

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