A319259 Minimal number of straight lines of a covering tree needed to cover n X n X n points arranged in a 3-D grid.

1, 4, 12, 20

Offset: 1

Author

Marco Ripà, Sep 15 2018

Keywords

• nonn
• hard
• more
• bref

Comments

For any n > 2, is it possible to construct an "inside the box" covering tree for any n X n X n set of points consisting of n^2 + n lines if n is even and n^2 + n + 1 lines if n is odd.

In the special case of any 2 X 2 X ... X 2 points problem (k-times n), every optimal covering tree has (exactly) 2^(k-1) rectilinear segments, thus 2^(3-1) = 4 lines for the 2 x 2 x 2 case.

If n = 3, then the (outside the box) solution is a(3) = 12, since such a connected arrangement of line segments has already been shown to exist, and this explicit upper bound matches 3^3 + 3. - Marco Ripà, Aug 25 2020

Let n >= 5, we know that it is impossible to cover more than n^3 points with n^2 segments (trivial), and we need at least n segments to obtain a connected graph (otherwise, we cannot join more than n + h*(n - 1) points with h + 1 connected lines). Thus, assuming n > 2, the general lower bound is confirmed to be n^2 + n, therefore a(n even) = 4, 20, 42, 72, ...

Examples

			For n = 3 the a(3) = 12 represents the minimum line segments to cover a 3 X 3 X 3 points (symmetrical) grid. - _Marco Ripà_, Aug 25 2020
		

Links

• A. Dumitrescu, C. D. Tóth, Covering Grids by Trees, 26th Canadian Conference on Computational Geometry, 2014.
• Marco Ripà, The rectangular spiral or the n1 X n2 X ... X nk Points Problem, Notes on Number Theory and Discrete Mathematics, 2014, 20(1), 59-71.
• Marco Ripà, Solving the 106 years old 3^k Points Problem with the Clockwise-algorithm, ResearchGate, 2020 (DOI: 10.13140/RG.2.2.34972.92802).

Crossrefs

Cf. A058992, A225227, A261547, A318165.

Formula

a(n) = n^2 + n for even n and for n = 3. - Marco Ripà, Aug 25 2020

Extensions

a(3) corrected by Marco Ripà, Aug 25 2020