On Classification of k-Dimension Paths in n-Cube

The shortest k-dimension paths (k-paths) between vertices of n-cube are considered on the basis a bijective mapping of k-faces into words over a finite alphabet. The presentation of such paths is proposed as (nk + 1)×n matrix of characters from the same alphabet. A classification of the paths is founded on numerical invariant as special partition. The partition consists of n parts, which correspond to columns of the matrix.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Ryabov, G. and Serov, V. (2014) On Classification of k-Dimension Paths in n-Cube. Applied Mathematics, 5, 723-727. doi: 10.4236/am.2014.54069.

 [1] Mollard, M. and Ramras, M. (2013) Edge Decompositions of Hypercubes by Paths and by Cycles. http://arxiv.org/pdf/1205.4161.pdf [2] Mundici, D. (2012) Logic on the n-Cube. http://arxiv.org/pdf/1207.5717.pdf [3] Leader, I. and Long, E. (2013) Long Geodesics in Subgraphs of the Cube. http://arxiv.org/pdf/1301.2195.pdf [4] Erde, J. (2013) Decomposing the Cube into Paths. http://xxx.tau.ac.il/pdf/1310.6776.pdf [5] Rota, G.-C. and Metropolis, N. (1978) Combinatorial Structure of the Faces of the n-Cube. SIAM Journal on Applied Mathematics, 35, 689-694. http://dx.doi.org/10.1137/0135057 [6] Stanley, R.P. (1999) Enumerative Combinatorics. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511609589 [7] Manin, Y.I. (1999) Classical Computing, Quantum Computing, and Shor’s Factoring Algorithm. http://arxiv.org/pdf/quant-ph/9903008.pdf [8] Ryabov, G.G. (2009) On the Quaternary Coding of Cubic Structures. (in Russian)http://num-meth.srcc.msu.ru/zhurnal/tom_2009/pdf/v10r138.pdf [9] Ryabov, G.G. (2011) Hausdorff Metric on Faces of the n-Cube. Journal of Mathematical Sciences, 177, 619-622. http://dx.doi.org/10.1007/s10958-011-0487-3