Kneser ranks of random graphs and minimum difference representations
Result 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 $0< p< 1$ there exist constants $c_i=c_i(p)>0$, $i=1,2$ such that with high probability [ c_1 n/(log n)< fKn(G) < 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 $c''_1, c''_2>0$ such that $c''_1 n/(log n)< f_{min}(G) < c''_2n/(log n)$ holds for almost all bipartite graphs $G$ on $n+n$ vertices.
Keywords
The result's identifiers
Result code in IS VaVaI
Result on the web
http://www.sciencedirect.com/science/article/pii/S1571065317301646?via%3Dihub
DOI - Digital Object Identifier
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 $0< p< 1$ there exist constants $c_i=c_i(p)>0$, $i=1,2$ such that with high probability [ c_1 n/(log n)< fKn(G) < 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 $c''_1, c''_2>0$ such that $c''_1 n/(log n)< f_{min}(G) < c''_2n/(log n)$ holds for almost all bipartite graphs $G$ on $n+n$ vertices.
Czech name
—
Czech description
—
Classification
Type
JSC - 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
GJ16-01602Y: Topological and geometric approaches to classes of permutations and graph properties
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
Basic information
Result type
JSC - Article in a specialist periodical, which is included in the SCOPUS database
OECD FORD
Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Year of implementation
2017