Edge Decompositions of Hypercubes by Paths and by Cycles

TL;DR

This paper investigates how hypercubes can be decomposed into paths and cycles, exploring conditions for divisibility and fundamental sets, and extends previous studies on graph divisibility within hypercubes.

Contribution

It advances the understanding of hypercube edge decompositions by analyzing paths, cycles, and fundamental sets, providing new results and generalizations.

Findings

Characterization of divisibility of hypercubes by paths and cycles

Conditions for the existence of fundamental sets in hypercubes

New theorems on edge decompositions of hypercubes

Abstract

If $H$ is (or is isomorphic to) a subgraph of $G$, $H$ is said to {\it divide} $G$ if there is an edge-decomposition of $G$ by copies of $E(H)$, the edge set of $H$. A more restrictive version of this is when there is a subgroup ${\cal H}$ of {\rm Aut} $(G)$, the automorphism group of $G$, such that the copies of $E(H)$ are the translates of $E(H)$ by the elements of ${\cal H}$. In a paper by the second author, this situation was described by saying that $H$, or more precisely $E(H)$, is a {\it fundamental} set for $G$. Many authors have studied the notion of divisibility for various graphs, and in particular for various subgraphs of hypercubes, such as paths, trees, and cycles. We continue such a study in this paper; both for divisibilty, and, when possible, for fundamental sets. The final section of the paper lists our main results.