On random irregular subgraphs

Random Structures and Algorithms Pub Date : 2023-11-28 DOI:10.1002/rsa.21204

Jacob Fox, Sammy Luo, Huy Tuan Pham

引用次数: 0

Abstract

Let

G $$ G $$

be a

d $$ d $$

-regular graph on

n $$ n $$

vertices. Frieze, Gould, Karoński, and Pfender began the study of the following random spanning subgraph model

H = H (G) $$ H=H(G) $$

. Assign independently to each vertex

v $$ v $$

G $$ G $$

a uniform random number

x (v) \in [0, 1] $$ x(v)\in \left[0,1\right] $$

, and an edge

(u, v) $$ \left(u,v\right) $$

G $$ G $$

is an edge of

H $$ H $$

if and only if

x (u) + x (v) \geq 1 $$ x(u)+x(v)\ge 1 $$

. Addressing a problem of Alon and Wei, we prove that if

d = o (n / {(\log n)}^{12}) $$ d=o\left(n/{\left(\log n\right)}^{12}\right) $$

, then with high probability, for each nonnegative integer

k \leq d $$ k\le d $$

, there are

(1 + o (1)) n / (d + 1) $$ \left(1+o(1)\right)n/\left(d+1\right) $$

vertices of degree

k $$ k $$

H $$ H $$

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在随机不规则子图上

设G $$ G $$是一个有n $$ n $$个顶点的d $$ d $$正则图。Frieze, Gould, Karoński和Pfender开始研究以下随机生成子图模型H=H(G) $$ H=H(G) $$。为G $$ G $$的每个顶点v $$ v $$独立分配一个均匀随机数x(v)∈[0,1]$$ x(v)\in \left[0,1\right] $$，并且当且仅当x(u)+x(v)≥1 $$ x(u)+x(v)\ge 1 $$时，G $$ G $$的边(u,v) $$ \left(u,v\right) $$就是H $$ H $$的边。针对Alon和Wei的问题，我们证明了如果d=o(n/(logn)12) $$ d=o\left(n/{\left(\log n\right)}^{12}\right) $$，那么对于每一个k≤d $$ k\le d $$的非负整数，H $$ H $$中有(1+o(1))n/(d+1) $$ \left(1+o(1)\right)n/\left(d+1\right) $$个k $$ k $$度的顶点。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Random Structures and Algorithms

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