European Journal of Combinatorics· 2026Q1
On the rigidity of random graphs in high-dimensional spaces
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- 2026year
Short summary
Random graphs ($G(n,p)$) exhibit distinct rigidity behaviors in high dimensions, separated by a critical probability $p_c = C_* rac{ ext{log } n}{n}$ where $C_* = rac{2}{1- ext{log } 2}$. Below this threshold, rigidity is dictated by the minimum degree; above it, rigidity scales with $np$.
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Key points
- Random graph rigidity in high dimensions depends on edge probability $p$.
- A critical probability $p_c = C_* rac{ ext{log } n}{n}$ separates two rigidity regimes.
- For $p < (1- ext{ε})p_c$, $d$-rigidity equals the minimum degree.
- For $p_c ext{ ≤ } p = o(n^{-1/2})$, $d$-rigidity equals $(1/2 + o(1))np$.
- The second regime confirms a conjecture by Krivelevich, Lew, and Michaeli.
AI-generated from the title and abstract; the full text is not read.
Abstract
We study the maximum dimension $d=d(n,p)$ for which an Erdős-Rényi $G(n,p)$ random graph is $d$-rigid. Our main results reveal two different regimes of rigidity in $G(n,p)$ separated at $p_c=C_*\log n/n,~C_*=2/(1-\log 2)$ -- the point where the graph's minimum degree exceeds half its average degree. We show that if $p < (1-\varepsilon)p_c $, then $d(n,p)$ is asymptotically almost surely (a.a.s.) equal to the minimum degree of $G(n,p)$. In contrast, if $p_c \leq p = o(n^{-1/2}) $ then $d(n,p) $ is a.a.s. equal to $(1/2 + o(1))np$. The second result confirms, in this regime, a conjecture of Krivelevich, Lew, and Michaeli.
The authors' abstract, as published at the source. European Journal of Combinatorics, 2026 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics