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Mathematics > Combinatorics

arXiv:1804.09029 (math)
[Submitted on 24 Apr 2018 (v1), last revised 13 Oct 2020 (this version, v2)]

Title:The $Q_2$-free process in the hypercube

Authors:J. Robert Johnson, Trevor Pinto
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Abstract:The generation of a random triangle-saturated graph via the triangle-free process has been studied extensively. In this short note our aim is to introduce an analogous process in the hypercube. Specifically, we consider the $Q_2$-free process in $Q_d$ and the random subgraph of $Q_d$ it generates. Our main result is that with high probability the graph resulting from this process has at least $cd^{2/3} 2^d$ edges. We also discuss a heuristic argument based on the differential equations method which suggests a stronger conjecture, and discuss the issues with making this rigorous. We conclude with some open questions related to this process.
Comments: 12 pages. Minor changes in response to referee's comments. To appear in The Electronic Journal of Combinatorics
Subjects: Combinatorics (math.CO); Probability (math.PR)
MSC classes: 05C35, 05C80
Cite as: arXiv:1804.09029 [math.CO]
  (or arXiv:1804.09029v2 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.1804.09029

Submission history

From: Robert Johnson [view email]
[v1] Tue, 24 Apr 2018 13:40:47 UTC (9 KB)
[v2] Tue, 13 Oct 2020 09:49:41 UTC (10 KB)
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