Tomáš Dvořák ; Petr Gregor ; Václav Koubek - Spanning paths in hypercubes

dmtcs:3442 - Discrete Mathematics & Theoretical Computer Science, January 1, 2005, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05) - https://doi.org/10.46298/dmtcs.3442
Spanning paths in hypercubesArticle

Authors: Tomáš Dvořák 1; Petr Gregor 1; Václav Koubek 2

  • 1 Faculty of Mathematics and Physics [Praha/Prague]
  • 2 Institute for Theoretical Computer Science

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.


Volume: DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Section: Proceedings
Published on: January 1, 2005
Imported on: May 10, 2017
Keywords: Hamiltonian paths,spanning paths,hypercube,vertex fault tolerance,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]

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