Conflict-Free Coloring of Intersection Graphs

Fekete, Sándor P.; Keldenich, Phillip

doi:10.1142/s0218195918500085

Cited by 9 publications

(4 citation statements)

References 28 publications

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“…Abel et al [1] studied a related notion of closed CF-coloring of planar graphs. Fekete and Keldenich [15] studied intersection graphs of unit discs and of unit squares, that is, graphs whose vertices are the geometric objects, and two objects are adjacent if their intersection is non-empty. Keller and Smorodinsky [26] studied intersection graphs G of pseudo-discs (i.e., simple Jordan regions such that the boundaries of any two such regions intersect in at most two points) and proved the asymptotically tight bound χ cf (G) = O(log n).…”

Section: Introduction 1backgroundmentioning

confidence: 99%

Conflict-Free Coloring of String Graphs

Keller

Rok

Smorodinsky

2020

Discrete Comput Geom

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Conflict-free coloring (in short, CF-coloring) of a graph G = (V, E) is a coloring of V such that the neighborhood of each vertex contains a vertex whose color differs from the color of any other vertex in that neighborhood. Bounds on CF-chromatic numbers have been studied both for general graphs and for intersection graphs of geometric shapes. In this paper we obtain such bounds for several classes of string graphs, i.e., intersection graphs of curves in the plane:(i) We provide a general upper bound of O(χ(G) 2 log n) on the CF-chromatic number of any string graph G with n vertices in terms of the classical chromatic number χ(G). This result stands in contrast to general graphs where the CF-chromatic number can be Ω( √ n) already for bipartite graphs.(ii) For some central classes of string graphs, the CF-chromatic number is as large as Θ( √ n),which was shown to be the upper bound for any graph even in the non-geometric context. For several such classes (e.g., intersection graphs of frames) we prove a tight bound of Θ(log n) with respect to the relaxed notion of k-CF-coloring (in which the punctured neighborhood of each vertex contains a color that appears at most k times), for a small constant k.(iii) We obtain a general upper bound on the k-CF-chromatic number of arbitrary hypergraphs: Any hypergraph with m hyperedges can be k-CF colored withÕ(m 1 k+1 ) colors. This bound, which extends a bound of Pach and Tardos (2009), is tight for some string graphs, up to a logarithmic factor.(iv) Our fourth result concerns circle graphs in which coloring problems are motivated by VLSI designs. We prove a tight bound of Θ(log n) on the CF-chromatic number of circle graphs, and an upper bound of O(log 3 n) for a wider class of string graphs that contains circle graphs, namely, intersection graphs of grounded L-shapes.

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Section: Introduction 1backgroundmentioning

confidence: 99%

Conflict-Free Coloring of String Graphs

Keller

Rok

Smorodinsky

2020

Discrete Comput Geom

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“…The problems are fixed parameter tractable when parameterized by vertex cover number, neighborhood diversity [4], distance to cluster [5], and more recently, treewidth [6,7]. This problem has attracted special interest for graphs arising out of intersection of geometric objects, see for instance, [8,9,10].…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Bounds for Conflict-free Coloring on Open Neighborhoods

Bhyravarapu¹,

Kalyanasundaram²

2020

Preprint

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In an undirected graph G, a conflict-free coloring with respect to open neighborhoods (denoted by CFON coloring) is an assignment of colors to the vertices such that every vertex has a uniquely colored vertex in its open neighborhood. The minimum number of colors required for a CFON coloring of G is the CFON chromatic number of G, denoted by χON (G). The decision problem that asks whether χON (G) ≤ k is NP-complete. Structural as well as algorithmic aspects of this problem have been well studied. We obtain the following results for χON (G):-Bodlaender, Kolay and Pieterse [WADS 2019] showed the upper bound χON (G) ≤ fvs(G) + 3, where fvs(G) denotes the size of a minimum feedback vertex set of G. We show the improved bound of χON (G) ≤ fvs(G) + 2, which is tight, thereby answering an open question in the above paper. -We study the relation between χON (G) and the pathwidth of the graph G, denoted pw(G). The above paper from WADS 2019 showed the upper bound χON (G) ≤ 2tw(G) + 1 where tw(G) stands for the treewidth of G. This implies an upper bound of χON (G) ≤ 2pw(G)+ 1. We show an improved bound of χON (G) ≤ ⌊ 5 3 (pw(G) + 1)⌋. -We prove new bounds for χON (G) with respect to the structural parameters neighborhood diversity and distance to cluster, improving the existing results of Gargano and Rescigno [Theor. Comput. Sci. 2015] and Reddy [Theor. Comput. Sci. 2018], respectively. Furthermore, our techniques also yield improved bounds for the closed neighborhood variant of the problem. -We also study the partial coloring variant of the CFON coloring problem, which allows vertices to be left uncolored. Let χ * ON (G) denote the minimum number of colors required to color G as per this variant. Abel et. al. [SIDMA 2018] showed that χ *ON (G) ≤ 8 when G is planar. They asked if fewer colors would suffice for planar graphs. We answer this question by showing that χ * ON (G) ≤ 5 for all planar G. This approach also yields the bound χ * ON (G) ≤ 4 for all outerplanar G. All our bounds are a result of constructive algorithmic procedures.

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“…In [18] intersection (and also inclusion and reverse-inclusion) hypergraphs of intervals of the line were considered. In [13] and [10] they considered intersection hypergraphs (and graphs) of (unit) disks, pseudo-disks, squares and axis-parallel rectangles.…”

Section: Introductionmentioning

confidence: 99%

Coloring Intersection Hypergraphs of Pseudo-Disks

Keszegh

2019

Discrete Comput Geom

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We prove that the intersection hypergraph of a family of n pseudo-disks with respect to another family of pseudo-disks admits a proper coloring with 4 colors and a conflict-free coloring with O(log n) colors. Along the way we prove that the respective Delaunay-graph is planar. We also prove that the intersection hypergraph of a family of n regions with linear union complexity with respect to a family of pseudo-disks admits a proper coloring with constantly many colors and a conflict-free coloring with O(log n) colors. Our results serve as a common generalization and strengthening of many earlier results, including ones about proper and conflict-free coloring points with respect to pseudo-disks, coloring regions of linear union complexity with respect to points and coloring disks with respect to disks.

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Conflict-Free Coloring of Intersection Graphs

Cited by 9 publications

References 28 publications

Conflict-Free Coloring of String Graphs

Conflict-Free Coloring of String Graphs

Combinatorial Bounds for Conflict-free Coloring on Open Neighborhoods

Coloring Intersection Hypergraphs of Pseudo-Disks

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