Sandwiching Biregular Random Graphs
Result description
Let G(n1,n2,m) be a uniformly random m-edge subgraph of the complete bipartite graph Kn1,n2 with bipartition (V1,V2) , where ni=|Vi| , i=1,2 . Given a real number p∈[0,1] such that d1:=pn2 and d2:=pn1 are integers, let R(n1,n2,p) be a random subgraph of Kn1,n2 with every vertex v∈Vi of degree di , i=1,2 . In this paper we determine sufficient conditions on n1,n2,p and m under which one can embed G(n1,n2,m) into R(n1,n2,p) and vice versa with probability tending to 1. In particular, in the balanced case n1=n2 , we show that if p≫logn/n and 1−p≫(logn/n)1/4 , then for some m∼pn2 , asymptotically almost surely one can embed G(n1,n2,m) into R(n1,n2,p) , while for p≫(log3n/n)1/4 and 1−p≫logn/n the opposite embedding holds. As an extension, we confirm the Kim–Vu Sandwich Conjecture for degrees growing faster than (nlogn)3/4 .
Keywords
Random regular graphBipartite random graphEmbedding CouplingMonotone graph property
The result's identifiers
Result code in IS VaVaI
Alternative codes found
RIV/00216208:11320/23:10476383
Result on the web
DOI - Digital Object Identifier
Alternative languages
Result language
angličtina
Original language name
Sandwiching Biregular Random Graphs
Original language description
Let G(n1,n2,m) be a uniformly random m-edge subgraph of the complete bipartite graph Kn1,n2 with bipartition (V1,V2) , where ni=|Vi| , i=1,2 . Given a real number p∈[0,1] such that d1:=pn2 and d2:=pn1 are integers, let R(n1,n2,p) be a random subgraph of Kn1,n2 with every vertex v∈Vi of degree di , i=1,2 . In this paper we determine sufficient conditions on n1,n2,p and m under which one can embed G(n1,n2,m) into R(n1,n2,p) and vice versa with probability tending to 1. In particular, in the balanced case n1=n2 , we show that if p≫logn/n and 1−p≫(logn/n)1/4 , then for some m∼pn2 , asymptotically almost surely one can embed G(n1,n2,m) into R(n1,n2,p) , while for p≫(log3n/n)1/4 and 1−p≫logn/n the opposite embedding holds. As an extension, we confirm the Kim–Vu Sandwich Conjecture for degrees growing faster than (nlogn)3/4 .
Czech name
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Czech description
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Classification
Type
Jimp - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Combinatorics Probability & Computing
ISSN
0963-5483
e-ISSN
1469-2163
Volume of the periodical
32
Issue of the periodical within the volume
1
Country of publishing house
GB - UNITED KINGDOM
Number of pages
44
Pages from-to
1-44
UT code for WoS article
000806622900001
EID of the result in the Scopus database
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Result type
Jimp - Article in a specialist periodical, which is included in the Web of Science database
OECD FORD
Pure mathematics
Year of implementation
2023