On Vertex Partitions of Hypercubes by Isometric Trees

Michel Mollard (1)

1 IF - Institut Fourier

When $n=2^m-1$ M.Ramras proved, by a counting argument, that for any isometrically embedded tree $T$ on $n$ edges in $Q_n$ there exists a group of translations $G$ such that $\left\{g(T); g\in G \right\}$ is a vertex partition of $Q_n$. Considering a more general context we are able to give an explicit construction of $G$ and can construct non group vertex partitions by isometric trees. We extend also this problem to vertex partition of $Q_{n'}$ by translates of an isometrically embedded tree on $n=2^m-1$ edges for any $n'\geq n$

Mots clés: Graph; Perfect code; Hypercube; Vertex Partition; Tiling

Domaines: Combinatoire [math.CO]

Michel Mollard. On Vertex Partitions of Hypercubes by Isometric Trees. SIAM Journal on Discrete Mathematics, 2011, 25, pp.534-538. ⟨hal-00535683v2⟩