Spanning paths in hypercubes

Dvořák, Tomáš ; Gregor, Petr ; Koubek, Václav

Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05) (2005).

Résumé

Given a family $\{u_i,v_i\}_{i=1}^k$ of pairwise distinct vertices of the $n$-dimensional hypercube $Q_n$ such that the distance of $u_i$ and $v_i$ is odd and $k \leq n-1$, there exists a family $\{P_i\}_{i=1}^k$ of paths such that $u_i$ and $v_i$ are the endvertices of $P_i$ and $\{V(P_i)\}_{i=1}^k$ partitions $V(Q_n)$. This holds for any $n \geq 2$ with one exception in the case when $n=k+1=4$. On the other hand, for any $n \geq 3$ there exist $n$ pairs of vertices satisfying the above condition for which such a family of spanning paths does not exist. We suggest further generalization of this result and explore a relationship to the problem of hamiltonicity of hypercubes with faulty vertices.

Détail

DOI : 10.46298/dmtcs.3442