Nearly spanning cycle in the percolated hypercube

TL;DR

The paper resolves the conjecture that the percolated hypercube $Q^d_p$ contains long cycles in the sparse regime by showing that for any fixed $\varepsilon>0$, there is a $C(\varepsilon)$ with $p\ge C/d$ ensuring whp a cycle of length at least $(1-\varepsilon)2^d$ exists. The authors develop a layered, multi-ingredient strategy: cover middle layers with many short paths, merge these paths through reservoir-augmented connections using a DFS-based auxiliary graph and a tree-growing framework called Merge-Or-Grow (MOG), and finally stitch the pieces into a nearly spanning cycle, aided by a coupling with mixed percolation on the hypercube. They introduce auxiliary machinery such as a path-extension forest, a dedicated MOG algorithm, and an Alternative Growth Procedure (AGP) to control growth and merging across layers, achieving a quantitative bound with inverse-polynomial dependence on the tolerance parameter. Moreover, they extend the approach to a mixed-percolation setting $Q^d_p(\delta)$, concluding that a cycle of length at least $(1-\varepsilon)\delta 2^d$ exists whp, thereby broadening the Hamiltonicity-type phenomena known for high-dimensional random subgraphs. The work advances our understanding of long cycles in high-dimensional percolation and provides a versatile toolkit potentially applicable to other layered graphs with similar structural decompositions.

Abstract

Let $Q^d$ be the $d$-dimensional binary hypercube. We form a random subgraph $Q^d_p\subseteq Q^d$ by retaining each edge of $Q^d$ independently with probability $p$. We show that, for every constant $\varepsilon>0$, there exists a constant $C=C(\varepsilon)>0$ such that, if $p\ge C/d$, then with high probability $Q^d_p$ contains a cycle of length at least $(1-\varepsilon)2^d$. This confirms a long-standing folklore conjecture, stated in particular by Condon, Espuny Díaz, Girão, Kühn, and Osthus [Hamiltonicity of random subgraphs of the hypercube, Mem. Amer. Math. Soc. 305 (2024), No. 1534].

Figures (3)

• Figure 1: Illustration of two paths $P_1, P_2$ merging via a path of length 2 through $V_2$. In each of $P_1,P_2$, the initial and the terminal segments $P_i^*$ appear in brown. Note that some vertices in these segments, those on the dashed lines, do not lie in the merged path.
• Figure 2: In the upper part of the figure, we have a path found in DFS-Aux. The red circles are vertices in $A$, and the blue circles are vertices in $B$. The black edges correspond to edges in $M$. In the lower part of the figure, we have the corresponding path in $Q^d_p$. The paths appear in black, with their initial and terminal segments, which correspond to vertices of $A$, in red (and slightly thinner). The vertices in $B\subseteq V_2$ are in blue.
• Figure 3: Illustration of two paths, $P$ and $R$, merging through $v$ by using the path $X$ from the tree $T_{P}^+$ and the path $Y$ from the tree $T_{R}^-$. The new path is then $P'$, and we define $T_{P}^-,T_{R}^+$ to be its trees.

Theorems & Definitions (49)

• Conjecture 1.1: Conjecture 1.3 of CDGKO21
• Theorem 1
• Remark 1
• Conjecture 1.2
• Conjecture 1.3
• Theorem 2.1: Lovász' version of the Kruskal-Katona theorem
• Lemma 2.2
• proof
• Lemma 2.3
• proof