Skip to main content
We gratefully acknowledge support from the Simons Foundation, member institutions, and all contributors. Donate
arXiv logo
Cornell University Logo

Mathematics > Combinatorics

arXiv:math/0702370 (math)
[Submitted on 13 Feb 2007 (v1), last revised 14 Oct 2008 (this version, v2)]

Title:Minimal percolating sets in bootstrap percolation

Authors:Robert Morris
View PDF
Abstract: In standard bootstrap percolation, a subset A of the n x n grid is initially infected. A new site is then infected if at least two of its neighbours are infected, and an infected site stays infected forever. The set A is said to percolate if eventually the entire grid is infected. A percolating set is said to be minimal if none of its subsets percolate. Answering a question of Bollobas, we show that there exists a minimal percolating set of size 4n^2/33 + o(n^2), but there does not exist one larger than (n + 2)^2/6.
Comments: 19 pgs, 4 figures, thorough rewrite, to appear in Electronic J. Combinatorics
Subjects: Combinatorics (math.CO)
Cite as: arXiv:math/0702370 [math.CO]
  (or arXiv:math/0702370v2 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.math/0702370

Submission history

From: Robert Morris [view email]
[v1] Tue, 13 Feb 2007 15:38:24 UTC (15 KB)
[v2] Tue, 14 Oct 2008 20:52:58 UTC (16 KB)
Full-text links:

Access Paper:

view license
Current browse context:
math.CO
< prev   |   next >
new | recent | 2007-02

References & Citations

export BibTeX citation

Bookmark

BibSonomy logo Reddit logo

Bibliographic and Citation Tools

Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)

Code, Data and Media Associated with this Article

alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)

Demos

Replicate (What is Replicate?)
Hugging Face Spaces (What is Spaces?)
TXYZ.AI (What is TXYZ.AI?)

Recommenders and Search Tools

Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)