Sandwiching biregular random graphs
Tereza Klimo\v{s}ov\'a, Christian Reiher, Andrzej Ruci\'nski, Matas, \v{S}ileikis

TL;DR
This paper investigates conditions under which random bipartite graphs with fixed degrees can be embedded into each other, confirming a conjecture for certain degree growth rates.
Contribution
It establishes sufficient conditions for embedding random bipartite graphs with fixed degrees into random subgraphs, confirming the Kim--Vu Sandwich Conjecture for faster-growing degrees.
Findings
Embedding conditions depend on edge probability p and degree parameters.
In the balanced case, embeddings occur with high probability under specified growth conditions.
Confirmed the Kim--Vu Sandwich Conjecture for degrees exceeding (n log n)^{3/4}.
Abstract
Let be a uniformly random -edge subgraph of the complete bipartite graph with bipartition , where . Given a real number such that and are integers, let be a random subgraph of such that every has degree , for . In this paper we determine sufficient conditions on , and under which one can embed into and vice versa with probability tending to . In particular, in the balanced case , we show that if and , then for some , asymptotically almost surely one can embed into , while for and we have the opposite embedding. As an extension, we confirm the Kim--Vu…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Stochastic processes and statistical mechanics
