Sandwiching Biregular Random Graphs

0558488 - ÚI 2024 RIV GB eng J - Journal Article
Klimošová, T. - Reiher, C. - Rucinski, A. - Šileikis, Matas
Sandwiching Biregular Random Graphs.
Combinatorics Probability & Computing. Roč. 32, č. 1 (2023), s. 1-44. ISSN 0963-5483. E-ISSN 1469-2163
R&D Projects: GA ČR(CZ) GJ20-27757Y
Institutional support: RVO:67985807
Keywords : Random regular graph * Bipartite random graph * Embedding Coupling * Monotone graph property
OECD category: Pure mathematics
Impact factor: 0.9, year: 2022
Method of publishing: Open access
https://doi.org/10.1017/S0963548322000049

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 .
Permanent Link: http://hdl.handle.net/11104/0332129