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

2018 ◽  
Vol 2020 (13) ◽  
pp. 4037-4061 ◽  
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.

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.

scholarly journals Coloring Chains for Compression with Uncertain Priors

10.37236/6468 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Noah Golowich
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Graph Coloring ◽  
Upper And Lower Bounds ◽  
Nice Property ◽  
Chromatic Numbers ◽  
Compression Scheme ◽  
Graph Coloring Problem ◽  
Close Relationship ◽  
Local Graph

Haramaty and Sudan considered the problem of transmitting a message between two people, Alice and Bob, when Alice's and Bob's priors on the message are allowed to differ by at most a given factor. To find a deterministic compression scheme for this problem, they showed that it is sufficient to obtain an upper bound on the chromatic number of a graph, denoted $U(N,s,k)$ for parameters $N,s,k$, whose vertices are nested sequences of subsets and whose edges are between vertices that have similar sequences of sets. In turn, there is a close relationship between the problem of determining the chromatic number of $U(N,s,k)$ and a local graph coloring problem considered by Erdős et al. We generalize the results of Erdős et al. by finding bounds on the chromatic numbers of graphs $H$ and $G$ when there is a homomorphism $\phi :H\rightarrow G$ that satisfies a nice property. We then use these results to improve upper and lower bounds on $\chi(U(N,s,k))$. 

Disjointness graphs of segments in the space

2020 ◽  
pp. 1-15
Author(s):  
János Pach ◽  
Gábor Tardos ◽  
Géza Tóth
Keyword(s):  
Chromatic Number ◽  
Pairwise Disjoint ◽  
Clique Number ◽  
Upper Bounds ◽  
Efficient Algorithms ◽  
Np Hard ◽  
Chromatic Numbers ◽  
Lines In Space ◽  
Vertex Set

Abstract The disjointness graph G = G(𝒮) of a set of segments 𝒮 in ${\mathbb{R}^d}$ , $$d \ge 2$$ , is a graph whose vertex set is 𝒮 and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies $\chi (G) \le {(\omega (G))^4} + {(\omega (G))^3}$ , where ω(G) denotes the clique number of G. It follows that 𝒮 has Ω(n1/5) pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments. We show that computing ω(G) and χ(G) for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colourings of G in which the number of colours satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free (ω(G) = 2), but whose chromatic numbers are arbitrarily large.

scholarly journals On The Locating-Chromatic Numbers of Subdivisions of Friendship Graph

2020 ◽  
Vol 26 (2) ◽  
pp. 175-184
Author(s):  
Brilly Maxel Salindeho ◽  
Hilda Assiyatun ◽  
Edy Tri Baskoro
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Connected Graph ◽  
Common Vertex ◽  
Chromatic Numbers

Let c be a k-coloring of a connected graph G and let pi={C1,C2,...,Ck} be the partition of V(G) induced by c. For every vertex v of G, let c_pi(v) be the coordinate of v relative to pi, that is c_pi(v)=(d(v,C1 ),d(v,C2 ),...,d(v,Ck )), where d(v,Ci )=min{d(v,x)|x in Ci }. If every two vertices of G have different coordinates relative to pi, then c is said to be a locating k-coloring of G. The locating-chromatic number of G, denoted by chi_L (G), is the least k such that there exists a locating k-coloring of G. In this paper, we determine the locating-chromatic numbers of some subdivisions of the friendship graph Fr_t, that is the graph obtained by joining t copies of 3-cycle with a common vertex, and we give lower bounds to the locating-chromatic numbers of few other subdivisions of Fr_t.

scholarly journals CUTTING A PART FROM MANY MEASURES

2019 ◽  
Vol 7 ◽  
Author(s):  
PAVLE V. M. BLAGOJEVIĆ ◽  
NEVENA PALIĆ ◽  
PABLO SOBERÓN ◽  
GÜNTER M. ZIEGLER
Keyword(s):  
Lower Bounds ◽  
Pairwise Disjoint ◽  
Positive Measure ◽  
Additional Property ◽  
Convex Hulls ◽  
Absolutely Continuous ◽  
Continuous Analogue ◽  
Finite Set ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

Holmsen, Kynčl and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^{d}$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset contains points of at least $d$ different colors, then there exists such a partition of $X$ with the additional property that the convex hulls of the $n$ subsets are pairwise disjoint.We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least $c$ different colors, where we also allow $c$ to be greater than $d$ . Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from $c$ different colors. For example, when $n\geqslant 2$ , $d\geqslant 2$ , $c\geqslant d$ with $m\geqslant n(c-d)+d$ are integers, and $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$ are $m$ positive finite absolutely continuous measures on $\mathbb{R}^{d}$ , we prove that there exists a partition of $\mathbb{R}^{d}$ into $n$ convex pieces which equiparts the measures $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{d-1}$ , and in addition every piece of the partition has positive measure with respect to at least $c$ of the measures $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$ .

scholarly journals The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

2021 ◽  
Author(s):  
Anne Driemel ◽  
André Nusser ◽  
Jeff M. Phillips ◽  
Ioannis Psarros
Keyword(s):  
Density Estimation ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Similarity Metrics ◽  
Vc Dimension ◽  
Set Systems ◽  
Polygonal Curves ◽  
The Individual ◽  
Curve Similarity ◽  
Metric Balls

AbstractThe Vapnik–Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and density estimation through the use of sampling bounds. We analyze set systems where the ground set X is a set of polygonal curves in $$\mathbb {R}^d$$ R d and the sets $$\mathcal {R}$$ R are metric balls defined by curve similarity metrics, such as the Fréchet distance and the Hausdorff distance, as well as their discrete counterparts. We derive upper and lower bounds on the VC dimension that imply useful sampling bounds in the setting that the number of curves is large, but the complexity of the individual curves is small. Our upper and lower bounds are either near-quadratic or near-linear in the complexity of the curves that define the ranges and they are logarithmic in the complexity of the curves that define the ground set.

On the Babai and Upper Chromatic Numbers of Graphs of Diameter 2

2021 ◽  
Vol 52 (1) ◽  
pp. 113-123
Author(s):  
Peter Johnson ◽  
Alexis Krumpelman
Keyword(s):  
Chromatic Number ◽  
Metric Space ◽  
Simple Graph ◽  
Chromatic Numbers

The Babai numbers and the upper chromatic number are parameters that can be assigned to any metric space. They can, therefore, be assigned to any connected simple graph. In this paper we make progress in the theory of the first Babai number and the upper chromatic number in the simple graph setting, with emphasis on graphs of diameter 2.

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