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arXiv · 1804.09029

The $Q_2$-free process in the hypercube

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.

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J. Robert Johnson, Trevor Pinto. 2018-04-24. The $Q_2$-free process in the hypercube. https://arxiv.org/abs/1804.09029

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