On 2-colored graphs and partitions of boxes

Holzman, Ron

doi:10.1016/j.ejc.2019.03.003

Cited by 3 publications

(1 citation statement)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that in three dimensions, the only types of intersection properties for brick partitions are kpiercing and k-slicing. Other types of intersection properties exist for higher dimensions, as well as for other types of partitions such as box partitions; we direct the interested readers to [1][2][3][4][5][6][7][8] for more results of this flavor.…”

Section: Introductionmentioning

confidence: 99%

Brick partition problems in three dimensions

Choi

Kim

Seo³

2021

Preprint

View full text Add to dashboard Cite

A d-dimensional brick is a set I 1 × • • • × I d where each I i is an interval. Given a brick B, a brick partition of B is a partition of B into bricks. A brick partition P d of a d-dimensional brick is kpiercing if every axis-parallel line intersects at least k bricks in P d . Bucic et al. [3] explicitly asked the minimum size p(d, k) of a k-piercing brick partition of a d-dimensional brick. The answer is known to be 4(k − 1) when d = 2. Our first result almost determines p(3, k). Namely, we construct a k-piercing brick partition of a 3-dimensional brick with 12k − 15 parts, which is off by only 1 from the known lower bound. As a generalization of the above question, we also seek the minimum size s(d, k) of a brick partition P d of a d-dimensional brick where each axis-parallel plane intersects at least k bricks in P d . We resolve the question in the 3-dimensional case by determining s(3, k) for all k.

show abstract