Coloring Graphs with Sparse Neighborhoods

The choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). It is shown that the choice number of the random graph G(n, p(n)) is almost surely Θ(ln(np(n)) ) whenever 2 < np(n) ≤ n/2. A related result for pseudo-random graphs is proved as well. By a special case of this result, the choice number (as well as the chromatic number) of any graph on n vertices with minimum degree at least n/2 − n 0.99 in which no two distinct vertices have more than n/4 + n 0.99 common neighbors is at most O(n/ ln n).

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“…We omit the detailed proof. Some extensions of this result have recently been proved in [5] and [23].…”

Section: Separated Eigenvalues and Sparse Neighborhoodsmentioning

confidence: 66%

List Coloring of Random and Pseudo-Random Graphs

1999

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“…Another proof is divided into two cases: every graph with maximum degree at most √ n has independence number Ω( √ n); on the other hand, if some vertex in the Delaunay graph has degree exceeding √ n, then its neighborhood would contain a monotone chain 1 All rectangles and boxes are axis-parallel in this paper. 2 The O or Ω notation hides log O(1) n and log O(1) m factors in this paper.…”

Section: Main Results On Points Wrt Rectangles In Two Dimensionsmentioning

confidence: 99%

“…This second proof can be refined to yield a slight improvement cf-color(n, B 2 ) = O( n/ log n) by using better bounds on the independence number of graphs with sparse neighborhoods [2], as several researchers have independently observed [19,24].…”

Section: Main Results On Points Wrt Rectangles In Two Dimensionsmentioning

confidence: 99%

Conflict-free coloring of points with respect to rectangles and approximation algorithms for discrete independent set

Chan

2012

Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry

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In the conflict-free coloring problem, for a given range space, we want to bound the minimum value F (n) such that every set P of n points can be colored with F (n) colors with the property that every nonempty range contains a unique color. We prove a new upper bound O(n 0.368 ) with respect to orthogonal ranges in two dimensions (i.e., axis-parallel rectangles), which is the first improvement over the previous bound O(n We also observe that combinatorial results on conflict-free coloring can be applied to the analysis of approximation algorithms for discrete versions of geometric independent set problems. Here, given a set P of (weighted) points and a set S of ranges, we want to select the largest(-weight) subset Q ⊂ P with the property that every range of S contains at most one point of Q. We obtain, for example, a randomized O(n 0.368 )-approximation algorithm for this problem with respect to orthogonal ranges in the plane.

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“…These proofs use a union bound over M = 2 Θ(N ) undesired events, by giving a 2 −Ω(N ) upper-bound on the probability of each of these events. 4 Unfortunately, there exist poly (log(N ))-wise independent graphs where any event that occurs with positive probability, has probability ≥ 2 −o(N ) . Therefore, directly 'de-randomizing' the original proof fails, and alternative arguments (suitable for the k-wise independent case) are provided.…”

Section: Our Techniques and Relations To Combinatorial Pseudorandomnessmentioning

confidence: 99%

“…More surprisingly, k = Θ(log(N )) suffices to capture a similar upper-bound (even for tiny densities p = c log(N )/N ). This upper-bound is based on Alon, Krivelevich and Sudakov's [3], [4] and on Johansson's [27].…”

Section: Coloring (See Section 55)mentioning

confidence: 99%

k-Wise Independent Random Graphs

Alon

Nussboim

2008

2008 49th Annual IEEE Symposium on Foundations of Computer Science

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We study the k-wise independent relaxation of the usual model G(N, p) of random graphs where, as in this model, N labeled vertices are fixed and each edge is drawn with probability p, however, it is only required that the distribution of any subset of k edges is independent. This relaxation can be relevant in modeling phenomena where only k-wise independence is assumed to hold, and is also useful when the relevant graphs are so huge that handling G(N, p) graphs becomes infeasible, and cheaper random-looking distributions (such as k-wise independent ones) must be used instead. Unfortunately, many well-known properties of random graphs in G(N, p) are global, and it is thus not clear if they are guaranteed to hold in the k-wise independent case. We explore the properties of k-wise independent graphs by providing upper-bounds and lower-bounds on the amount of independence, k, required for maintaining the main properties of G(N, p) graphs: connectivity, Hamiltonicity, the connectivitynumber, clique-number and chromatic-number and the appearance of fixed subgraphs. Most of these properties are shown to be captured by either constant k or by some k = poly (log(N )) for a wide range of values of p, implying that random looking graphs on N vertices can be generated by a seed of size poly(log(N )). The proofs combine combinatorial, probabilistic and spectral techniques.

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Coloring Graphs with Sparse Neighborhoods

Cited by 112 publications

References 7 publications

List Coloring of Random and Pseudo-Random Graphs

List Coloring of Random and Pseudo-Random Graphs

Conflict-free coloring of points with respect to rectangles and approximation algorithms for discrete independent set

k-Wise Independent Random Graphs

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