What is the pathwidth of the 3D-grid (mesh or lattice) with sidelength k?
The pathwidth of $P^3_k$ can be determined as a corollary to some known results.
FitzGerald [2] showed that the bandwidth of $P^3_k$ is $\lfloor \frac{3}{4} k^{2} + \frac{1}{2} k \rfloor$.
Harper [3] showed a condition such that if a graph satisfies the condition, then its pathwidth and bandwidth are the same.
Moghadam [4,5] and Bollobás and Leader [1] independently showed that any multi-dimensional grid satisfies Harper's condition.
These results imply that the pathwidth of $P^3_k$ is also $\lfloor \frac{3}{4} k^{2} + \frac{1}{2}k \rfloor$.

In our paper mentioned by Hsien-Chih, we generalized FitzGerald's result as Yoshio explained.
I believe the treewidth of $P^3_k$ is not known.

FYI: I've just submitted an English version of our paper to arXiv.


B. Bollobás and I. Leader, Compressions and isoperimetric inequalities, J. Combin. Theory Ser. A 56 (1991) 47-62.
C.H. FitzGerald, Optimal indexing of the vertices of graphs, Math. Comp. 28 (1974), 825-831.
L.H. Harper, Optimal numberings and isoperimetric problems on graphs, J. Combin. Theory 1 (1966) 385-393.
H.S. Moghadam, Compression operators and a solution to the bandwidth problem of the product of $n$ paths, Ph.D. thesis, University of California, Riverside (1983).
H.S. Moghadam, Bandwidth of the product of $n$ paths, Congr. Numer. 173 (2005) 3-15.