Clique Colourings of Geometric Graphs

McDiarmid, Colin; Mitsche, Dieter; Prałat, Paweł

doi:10.37236/7159

Cited by 3 publications

(3 citation statements)

References 26 publications

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“…On the algorithmic side, it is known that testing whether χ c (G) = 2 for a planar graph can be performed in polynomial time [16], but deciding whether χ c (G) = 2 is NP -complete for 3-chromatic perfect graphs [16] and for graphs with maximum degree 3 [2]. The clique chromatic number for geometric graphs (in particular, random geometric graphs) is analysed in the accompanying paper [22].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Clique coloring of binomial random graphs

McDiarmid

Mitsche

Prałat

2018

Random Struct Algorithms

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A clique coloring of a graph is a coloring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The smallest number of colors in such a coloring is the clique chromatic number. In this paper, we study the asymptotic behavior of the clique chromatic number of the random graph 𝒢(n,p) for a wide range of edge‐probabilities p = p(n). We see that the typical clique chromatic number, as a function of the average degree, forms an intriguing step function.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Clique coloring of binomial random graphs

McDiarmid

Mitsche

Prałat

2018

Random Struct Algorithms

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“…This was recently disproved in [CPTT16]. See [MMP16] for recent work on clique-colourings of binomial random graphs and of geometric graphs. We prove Theorem 5.1 using Theorem 2.3.…”

Section: Clique Colouringmentioning

confidence: 99%

Random perfect graphs

McDiarmid

Yolov

2018

Random Struct Algorithms

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We investigate the asymptotic structure of a random perfect graph Pn sampled uniformly from the set of perfect graphs on vertex set {1,…,n}. Our approach is based on the result of Prömel and Steger that almost all perfect graphs are generalised split graphs, together with a method to generate such graphs almost uniformly. We show that the distribution of the maximum of the stability number α(Pn) and clique number ω(Pn) is close to a concentrated distribution L(n) which plays an important role in our generation method. We also prove that the probability that Pn contains any given graph H as an induced subgraph is asymptotically 0 or 12 or 1. Further we show that almost all perfect graphs are 2‐clique‐colorable, improving a result of Bacsó et al. from 2004; they are almost all Hamiltonian; they almost all have connectivity κ(Pn) equal to their minimum degree; they are almost all in class one (edge‐colorable using Δ colors, where Δ is the maximum degree); and a sequence of independently and uniformly sampled perfect graphs of increasing size converges almost surely to the graphon WP(x,y)=12 (1[x≤1/2]+ 1[y≤1/2]).

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“…Conflict-free coloring is not the only type of coloring for which unit disk graphs have been found to require a bounded number of colors. In their recent work, McDiarmid et al [25] consider clique coloring of unit disk graphs, in particular with regard to the asymptotic behavior of the clique chromatic number of random unit disk graphs. They also prove that every unit disk graph in the plane can be colored with nine colors, while three colors are sometimes necessary.…”

Section: Introductionmentioning

confidence: 99%

Conflict-Free Coloring of Intersection Graphs

Fekete

Keldenich

2018

Int. J. Comput. Geom. Appl.

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A conflict-free k-coloring of a graph G = (V, E) 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 conflictfree coloring of geometric intersection graphs. We demonstrate that the intersection graph of n geometric objects without fatness properties and size restrictions may have conflict-free chromatic number in Ω(log n/ log log n) and in Ω( √ log n) for disks or squares of different sizes; it is known for general graphs that the worst case is in Θ(log 2 n). For unit-disk intersection graphs, we prove that it is NP-complete to decide the existence of a conflictfree coloring with one color; we also show that six colors always suffice, using an algorithm that colors unit disk graphs of restricted height with two colors. We conjecture that four colors are sufficient, which we prove for unit squares instead of unit disks. For interval graphs, we establish a tight worst-case bound of two.

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Clique Colourings of Geometric Graphs

Cited by 3 publications

References 26 publications

Clique coloring of binomial random graphs

Clique coloring of binomial random graphs

Random perfect graphs

Conflict-Free Coloring of Intersection Graphs

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