Combinatorics and Graph Theory

    

General Conjecture on the Optimal Covering Trails in a K-Dimensional Cubic Lattice

Authors: Marco Ripà

We introduce a general conjecture involving minimum-link covering trails for any given k-dimensional grid n × n × ··· × n, belonging to the cubic lattice ℕ^k. In detail, if n is above two, we hypothesize that the minimal link length of any covering trail, for the above-mentioned set of n^k points in the Euclidean space ℝ^k, is equal to h(n, k) = (n^k − 1)/(n − 1) + c·(n − 3), where c = k − 1 iff h(4, 3) = 23, c = 1 iff h(4, 3) = 22, or even c = 0 iff h(4, 3) = 21.

Comments: 7 Pages.

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Submission history

[v1] 2023-07-03 22:22:34
[v2] 2024-01-05 01:58:19

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