Measurable chromatic number of geometric graphs and sets without some distances in euclidean space

COMBINATORICA ◽  
1984 ◽  
Vol 4 (2-3) ◽  
pp. 213-218 ◽  
Author(s):  
L. A. Székely
Keyword(s):  
Euclidean Space ◽  
Chromatic Number ◽  
Geometric Graphs ◽  
Measurable Chromatic Number

scholarly journals Coloring Geographical Threshold Graphs

2010 ◽  
Vol Vol. 12 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Milan Bradonjic ◽  
Tobias Mueller ◽  
Allon G. Percus
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Chromatic Number ◽  
Clique Number ◽  
Threshold Function ◽  
The Internet ◽  
Geometric Graphs ◽  
Random Geometric Graphs ◽  
International Audience ◽  
Threshold Graph

Graphs and Algorithms International audience We propose a coloring algorithm for sparse random graphs generated by the geographical threshold graph (GTG) model, a generalization of random geometric graphs (RGG). In a GTG, nodes are distributed in a Euclidean space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. The motivation for analyzing this model is that many real networks (e. g., wireless networks, the Internet, etc.) need to be studied by using a ''richer'' stochastic model (which in this case includes both a distance between nodes and weights on the nodes). Here, we analyze the GTG coloring algorithm together with the graph's clique number, showing formally that in spite of the differences in structure between GTG and RGG, the asymptotic behavior of the chromatic number is identical: chi = ln n/ln ln n(1 +o(1)). Finally, we consider the leading corrections to this expression, again using the coloring algorithm and clique number to provide bounds on the chromatic number. We show that the gap between the lower and upper bound is within C ln n/(ln ln n)(2), and specify the constant C.

On the measurable chromatic number of a space of dimension n ≤ 24

2015 ◽  
Vol 92 (3) ◽  
pp. 761-763 ◽  
Author(s):  
L. I. Bogolubsky ◽  
A. M. Raigorodskii
Keyword(s):  
Chromatic Number ◽  
Measurable Chromatic Number

scholarly journals Clique Colourings of Geometric Graphs

10.37236/7159 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Colin McDiarmid ◽  
Dieter Mitsche ◽  
Pawel Prałat
Keyword(s):  
Asymptotic Behaviour ◽  
Chromatic Number ◽  
Euclidean Distance ◽  
High Probability ◽  
Maximal Clique ◽  
Higher Dimensions ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Vertex Set

A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number.  Given $n$ points $\mathbf{x}_1, \ldots,\mathbf{x}_n$ in the plane, and a threshold $r>0$, the corresponding geometric graph has vertex set $\{v_1,\ldots,v_n\}$, and distinct $v_i$ and $v_j$ are adjacent when the Euclidean distance between $\mathbf{x}_i$ and $\mathbf{x}_j$ is at most $r$. We investigate the clique chromatic number of such graphs.We first show that the clique chromatic number is at most 9 for any geometric graph in the plane, and briefly consider geometric graphs in higher dimensions. Then we study the asymptotic behaviour of the clique chromatic number for the random geometric graph $\mathcal{G}$ in the plane, where $n$ random points are independently and uniformly distributed in a suitable square. We see that as $r$ increases from 0, with high probability the clique chromatic number is 1 for very small $r$, then 2 for small $r$, then at least 3 for larger $r$, and finally drops back to 2.

scholarly journals Bounds on the measurable chromatic number of Rn

1989 ◽  
Vol 75 (1-3) ◽  
pp. 343-372 ◽  
Author(s):  
L.A. Székely ◽  
N.C. Wormald
Keyword(s):  
Chromatic Number ◽  
Measurable Chromatic Number

On the chromatic number of Euclidean space with two forbidden distances

2014 ◽  
Vol 96 (5-6) ◽  
pp. 827-830 ◽  
Author(s):  
A. V. Berdnikov ◽  
A. M. Raigorodskii
Keyword(s):  
Euclidean Space ◽  
Chromatic Number

scholarly journals Schrödinger operators with δ- and δ′-interactions on Lipschitz surfaces and chromatic numbers of associated partitions

2014 ◽  
Vol 26 (08) ◽  
pp. 1450015 ◽  
Author(s):  
Jussi Behrndt ◽  
Pavel Exner ◽  
Vladimir Lotoreichik
Keyword(s):  
Euclidean Space ◽  
Chromatic Number ◽  
Spectral Properties ◽  
Schrödinger Operators ◽  
Operator Inequality ◽  
Schrodinger Operators ◽  
Lipschitz Domains ◽  
Chromatic Numbers ◽  
Combinatorial Properties

We investigate Schrödinger operators with δ- and δ′-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result, we prove an operator inequality for the Schrödinger operators with δ- and δ′-interactions which is based on an optimal coloring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schrödinger operators and, in particular, it allows to transform known results for Schrödinger operators with δ-interactions to Schrödinger operators with δ′-interactions.

Measurable chromatic numbers

2008 ◽  
Vol 73 (4) ◽  
pp. 1139-1157 ◽  
Author(s):  
Benjamin D. Miller
Keyword(s):  
Chromatic Number ◽  
Equivalence Relation ◽  
Initial Segment ◽  
Polish Space ◽  
Borel Function ◽  
Chromatic Numbers ◽  
Unilateral Shift ◽  
Dichotomy Theorem ◽  
Borel Functions ◽  
Measurable Chromatic Number

AbstractWe show that if add(null) = c, then the globally Baire and universally measurable chromatic numbers of the graph of any Borel function on a Polish space are equal and at most three. In particular, this holds for the graph of the unilateral shift on [ℕ]ℕ, although its Borel chromatic number is ℵ0. We also show that if add(null) = c, then the universally measurable chromatic number of every treeing of a measure amenable equivalence relation is at most three. In particular, this holds for “the” minimum analytic graph with uncountable Borel (and Baire measurable) chromatic number. In contrast, we show that for all κ Є6 (2, 3…..ℵ0, c), there is a treeing of E0 with Borel and Baire measurable chromatic number κ. Finally, we use a Glimm–Effros style dichotomy theorem to show that every basis for a non-empty initial segment of the class of graphs of Borel functions of Borel chromatic number at least three contains a copy of (ℝ<ℕ, ⊇).

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