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arXiv · 2311.16631

Climbing up a random subgraph of the hypercube

Michael Anastos·Sahar Diskin·Dor Elboim·Michael Krivelevich
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Abstract

Let $Q^d$ be the $d$-dimensional binary hypercube. We say that $P=\{v_1,\ldots, v_k\}$ is an increasing path of length $k-1$ in $Q^d$, if for every $i\in [k-1]$ the edge $v_iv_{i+1}$ is obtained by switching some zero coordinate in $v_i$ to a one coordinate in $v_{i+1}$. Form a random subgraph $Q^d_p$ by retaining each edge in $E(Q^d)$ independently with probability $p$. We show that there is a phase transition with respect to the length of a longest increasing path around $p=\frac{e}{d}$. Let $\alpha$ be a constant and let $p=\frac{\alpha}{d}$. When $\alpha e$, whp there is a path of length $d-2$ in $Q^d_p$, and in fact, whether it is of length $d-2, d-1$, or $d$ depends on whether the all-zero and all-one vertices percolate or not.

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Michael Anastos, Sahar Diskin, Dor Elboim, Michael Krivelevich. 2023-11-28. Climbing up a random subgraph of the hypercube. https://arxiv.org/abs/2311.16631

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