On three measures on non-convexity

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F17%3A10360661" target="_blank" >RIV/00216208:11320/17:10360661 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1007/s11856-017-1467-1" target="_blank" >http://dx.doi.org/10.1007/s11856-017-1467-1</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s11856-017-1467-1" target="_blank" >10.1007/s11856-017-1467-1</a>

Result language
angličtina
Original language name
On three measures on non-convexity
Original language description
The invisibility graph I(X) of a set X in R^d is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in X. We consider the following three parameters of a set X: the clique number omega(I(X)), the chromatic number chi(I(X)) and the convexity number gamma(X), which is the minimum number of convex subsets of X that cover X. We settle a conjecture of Matoušek and Valtr claiming that for every planar set X, gamma(X) can be bounded in terms of chi(I(X)). As a part of the proof we show that a disc with n one-point holes near its boundary has chi(I(X)) >= log log(n) but omega(I(X))=3. We also find sets X in R^5 with chi(X)=2, but gamma(X) arbitrarily large.
Czech name
—
Czech description
—

Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GBP202%2F12%2FG061" target="_blank" >GBP202/12/G061: Center of excellence - Institute for theoretical computer science (CE-ITI)</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2017
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Similar results(10)