Estimate for the chromatic number of Euclidean space with several forbidden distances

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.

Download Full-text

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

COMBINATORICA ◽

10.1007/bf02579223 ◽

1984 ◽

Vol 4 (2-3) ◽

pp. 213-218 ◽

Cited By ~ 8

Author(s):

L. A. Székely

Keyword(s):

Euclidean Space ◽

Chromatic Number ◽

Geometric Graphs ◽

Measurable Chromatic Number

Download Full-text

On the chromatic number of Euclidean space and the Borsuk problem

Mathematical Notes ◽

10.1134/s0001434608030322 ◽

2008 ◽

Vol 83 (3-4) ◽

pp. 579-582 ◽

Cited By ~ 1

Author(s):

A. M. Raigorodskii ◽

I. M. Shitova

Keyword(s):

Euclidean Space ◽

Chromatic Number ◽

Borsuk Problem

Download Full-text

Coloring Geographical Threshold Graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.497 ◽

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.

Download Full-text

Chromatic Numbers of Stable Kneser Hypergraphs via Topological Tverberg-Type Theorems

International Mathematics Research Notices ◽

10.1093/imrn/rny135 ◽

2018 ◽

Vol 2020 (13) ◽

pp. 4037-4061 ◽

Cited By ~ 2

Author(s):

Florian Frick

Keyword(s):

Euclidean Space ◽

Chromatic Number ◽

Lower Bounds ◽

Pairwise Disjoint ◽

Convex Hulls ◽

Chromatic Numbers ◽

Set Systems ◽

Equivariant Topology ◽

Disjoint Sets ◽

Combination Of Methods

Abstract Kneser’s 1955 conjecture—proven by Lovász in 1978—asserts that in any partition of the $k$-subsets of $\{1, 2, \dots , n\}$ into $n-2k+1$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to significantly fewer $k$-sets and still observe the same intersection pattern. Alon, Frankl, and Lovász proved a different generalization of Kneser’s conjecture for $r$ pairwise disjoint sets. Dolnikov generalized Lovász’ result to arbitrary set systems, while Kříž did the same for the $r$-fold extension of Kneser’s conjecture. Here we prove a common generalization of all of these results. Moreover, we prove additional strengthenings by determining the chromatic number of certain sparse stable Kneser hypergraphs, and further develop a general approach to establishing lower bounds for chromatic numbers of hypergraphs using a combination of methods from equivariant topology and intersection results for convex hulls of points in Euclidean space.

Download Full-text