Edge Decompositions of Hypercubes by Paths and by Cycles

Michel Mollard (1) , Mark Ramras (2)

1 IF - Institut Fourier
2 neu - Departement of Mathematics [Boston]

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.

Mots clés: Hamiltonian cycle; Hypercube; edge-decomposition; path; cycle

Domaines: Combinatoire [math.CO]

Michel Mollard, Mark Ramras. Edge Decompositions of Hypercubes by Paths and by Cycles. Graphs and Combinatorics, 2015. ⟨hal-00698983v3⟩