Completely independent spanning trees in the hypercube

TL;DR

This work studies completely independent spanning trees in hypercubes and their Cartesian powers. By encoding vertices with a vector-space label and carefully alternating row/column constructions, it builds $2^k$ edge-disjoint spanning trees with disjoint interiors for large $n$, achieving $\Omega(n)$ such trees in $Q_n$ and diameter close to the lower bound $(2+o(1))n$. It further refines the construction to reduce diameter to $(2+o(1))n$ while maintaining linear growth in the number of trees, using a constant number of junction rows. The results extend to general Cartesian powers $H^n$, showing the same asymptotic number of trees and diameter scaling of $2n\,r(H)$, where $r(H)$ is the radius of $H$. The paper closes with open questions on the exact linear-constant and limit behavior of the maximum number of completely independent spanning trees in hypercubes.

Abstract

We say two spanning trees of a graph are completely independent if their edge sets are disjoint, and for each pair of vertices, the paths between them in each spanning tree do not have any other vertex in common. Pai and Chang constructed two such spanning trees in the hypercube $Q_n$ for sufficiently large $n$, while Kandekar and Mane recently showed there are $3$ pairwise completely independent spanning trees in hypercubes $Q_n$ for sufficiently large $n$. We prove that for each $k$, there exist $k$ completely independent spanning trees in $Q_n$ for sufficiently large $n$. In fact, we show that there are $(\frac{1}{12}+o(1))n$ spanning trees in $Q_n$, each with diameter $(2+o(1))n$. As the minimal diameter of any spanning tree of $Q_n$ is $2n-1$, this diameter is asymptotically optimal. We prove a similar result for the powers $H^n$ of any fixed graph $H$.

Figures (1)

• Figure 1: The structure of tree $T_\alpha$. Here vertices of $Q_n$ are represented as $(u;w)\in Q_m\times Q_{n-m}$. Vertices of label $\alpha$ are shown in green, with junctions of $T_\alpha$ indicated in red, and those of another $T_\beta$ in blue.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Theorem 3.1
• Claim 3.2
• proof
• Claim 3.3
• proof