KNESER RANKS OF RANDOM GRAPHS AND MINIMUM DIFFERENCE REPRESENTATIONS

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F18%3A10387686" target="_blank" >RIV/00216208:11320/18:10387686 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1137/17M1114703" target="_blank" >https://doi.org/10.1137/17M1114703</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1137/17M1114703" target="_blank" >10.1137/17M1114703</a>

Jazyk výsledku

angličtina

Název v původním jazyce

KNESER RANKS OF RANDOM GRAPHS AND MINIMUM DIFFERENCE REPRESENTATIONS

Popis výsledku v původním jazyce

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 -&gt; A(v) to the vertices v is an element of V such that A(u) and A(v) are disjoint if and only if uv is an element of E. The smallest such k is called the Kneser rank of G and denoted by fK(neser) (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 G is an element of G (n; p) satisfies with high probability c(1)n/(log n) &lt; fK(neser) (G) &lt; c(2)n= (log n) : We apply this for other graph representations defined by Boros, Gurvich, and Meshulam. A k-mindifference representation of a graph G is an assignment of a set A(i) to each vertex i is an element of V (G) such that ij is an element of E (G) double left right arrow min {vertical bar Ai A(i) vertical bar, vertical bar A(j) A(i vertical bar)vertical bar}&gt;= k: The smallest k such that there exists a k-mindi ff erence representation of G is denoted by fmin (G). Balogh and Prince proved in 2009 that for every k there is a graph G with f(min) (G) &gt;= 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;(2) n / (log n) holds for almost all bipartite graphs G on n + n vertices.

Název v anglickém jazyce

KNESER RANKS OF RANDOM GRAPHS AND MINIMUM DIFFERENCE REPRESENTATIONS

Popis výsledku anglicky

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 -&gt; A(v) to the vertices v is an element of V such that A(u) and A(v) are disjoint if and only if uv is an element of E. The smallest such k is called the Kneser rank of G and denoted by fK(neser) (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 G is an element of G (n; p) satisfies with high probability c(1)n/(log n) &lt; fK(neser) (G) &lt; c(2)n= (log n) : We apply this for other graph representations defined by Boros, Gurvich, and Meshulam. A k-mindifference representation of a graph G is an assignment of a set A(i) to each vertex i is an element of V (G) such that ij is an element of E (G) double left right arrow min {vertical bar Ai A(i) vertical bar, vertical bar A(j) A(i vertical bar)vertical bar}&gt;= k: The smallest k such that there exists a k-mindi ff erence representation of G is denoted by fmin (G). Balogh and Prince proved in 2009 that for every k there is a graph G with f(min) (G) &gt;= 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;(2) n / (log n) holds for almost all bipartite graphs G on n + n vertices.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

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

Projekt

<a href="/cs/project/GJ16-01602Y" target="_blank" >GJ16-01602Y: Topologické a geometrické přístupy k permutačním třídám a grafovým vlastnostem</a><br>

Návaznosti

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

Rok uplatnění

2018

Kód důvěrnosti údajů

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

Název periodika

SIAM Journal on Discrete Mathematics

ISSN

0895-4801

e-ISSN

—

Svazek periodika

32

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

13

Strana od-do

1016-1028

Kód UT WoS článku

000436975900014

EID výsledku v databázi Scopus

2-s2.0-85049599616