Uniform Partitions of 3-space, their Relatives and Embedding

Uniform Partitions of 3-space, their Relatives and Embedding

We review 28 uniform partitions of 3-space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic lattice Zn . We also consider some relatives of those 28 partitions, including Archimedean 4-polytopes of Conway–Guy, non-compact uniform partitions, Kelvin partitions and those with a unique vertex figure (i.e., Delaunay star). Among t...

J un 1 99 9 Uniform partitions of 3 - space , their relatives and embedding ∗

We review 28 uniform partitions of 3-space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic lattice Zn. We also consider some relatives of those 28 partitions, including Achimedean 4-polytopes of Conway-Guy, non-compact uniform partitions, Kelvin partitions and those with unique vertex figure (i.e. Delaunay star). Among last o...

. M G ] 6 J un 1 99 9 Uniform partitions of 3 - space , their relatives and embedding ∗

We review 28 uniform partitions of 3-space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic lattice Zn. We also consider some relatives of those 28 partitions, including Achimedean 4-polytopes of Conway-Guy, non-compact uniform partitions, Kelvin partitions and those with unique vertex figure (i.e. Delaunay star). Among last o...

Judicious Partitions of 3-uniform Hypergraphs

A conjecture of Bollobás and Thomason asserts that, for r ≥ 1, every r-uniform hypergraph with m edges can be partitioned into r classes such that every class meets at least rm/(2r−1) edges. Bollobás, Reed and Thomason [3] proved that there is a partition in which every edge meets at least (1 − 1/e)m/3 ≈ 0.21m edges. Our main aim is to improve this result for r = 3. We prove that every 3-unifor...

Embedding Factorizations for 3-Uniform Hypergraphs