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Kneser ranks of random graphs and minimum difference representations

The result's identifiers

  • Result code in IS VaVaI

    <a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F17%3A10367120" target="_blank" >RIV/00216208:11320/17:10367120 - isvavai.cz</a>

  • Result on the web

    <a href="http://www.sciencedirect.com/science/article/pii/S1571065317301646?via%3Dihub" target="_blank" >http://www.sciencedirect.com/science/article/pii/S1571065317301646?via%3Dihub</a>

  • DOI - Digital Object Identifier

    <a href="http://dx.doi.org/10.1016/j.endm.2017.06.079" target="_blank" >10.1016/j.endm.2017.06.079</a>

Alternative languages

  • Result language

    angličtina

  • Original language name

    Kneser ranks of random graphs and minimum difference representations

  • Original language description

    Every graph $G=(V,E)$ is an induced subgraph of some Kneser graph of rank $k$, i.e., there is an assignment of (distinct) $k$-sets $v mapsto A_v$ to the vertices $vin V$ such that $A_u$ and $A_v$ are disjoint if and only if $uvin E$. The smallest such $k$ is called the {em Kneser rank} of $G$ and denoted by $fKn(G)$. As an application of a result of Frieze and Reed concerning the clique cover number of random graphs we show that for constant $0&lt; p&lt; 1$ there exist constants $c_i=c_i(p)&gt;0$, $i=1,2$ such that with high probability [ c_1 n/(log n)&lt; fKn(G) &lt; c_2 n/(log n). ] We apply this to other graph representations defined by Boros, Gurvich and Meshulam. A {em $k$-min-difference representation} of a graph $G$ is an assignment of a set $A_i$ to each vertex $iin V(G)$ such that $ ijin E(G) ,, Leftrightarrow , , min {|A_isetminus A_j|,|A_jsetminus A_i| }geq k. $ The smallest $k$ such that there exists a $k$-min-difference representation of $G$ is denoted by $f_{min}(G)$. Balogh and Prince proved in 2009 that for every $k$ there is a graph $G$ with $f_{min}(G)geq k$. We prove that there are constants $c&apos;&apos;_1, c&apos;&apos;_2&gt;0$ such that $c&apos;&apos;_1 n/(log n)&lt; f_{min}(G) &lt; c&apos;&apos;_2n/(log n)$ holds for almost all bipartite graphs $G$ on $n+n$ vertices.

  • Czech name

  • Czech description

Classification

  • Type

    J<sub>SC</sub> - Article in a specialist periodical, which is included in the SCOPUS database

  • CEP classification

  • OECD FORD branch

    10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Result continuities

  • Project

    <a href="/en/project/GJ16-01602Y" target="_blank" >GJ16-01602Y: Topological and geometric approaches to classes of permutations and graph properties</a><br>

  • Continuities

    P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Others

  • Publication year

    2017

  • Confidentiality

    S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Data specific for result type

  • Name of the periodical

    Electronic Notes in Discrete Mathematics

  • ISSN

    1571-0653

  • e-ISSN

  • Volume of the periodical

    61

  • Issue of the periodical within the volume

    August

  • Country of publishing house

    NL - THE KINGDOM OF THE NETHERLANDS

  • Number of pages

    5

  • Pages from-to

    499-503

  • UT code for WoS article

  • EID of the result in the Scopus database

    2-s2.0-85026787337