Let d be a positive integer. A graph G is called d-divisible if d divides the degree of each vertex of G. G is called nowhere d-divisible if no degree of a vertex of G is divisible by d. For a graph H, gcd(H) denotes the greatest common divisor of the degrees of the vertices of H. The H-packing number of G is the maximum number of pairwise edge disjoint copies of H in G. The H-covering number o...

We study the Hard Core Model on the graphs G obtained from Archimedean tilings i.e. configurations in {0,1} with the nearest neighbor 1’s forbidden. Our particular aim in choosing these graphs is to obtain insight to the geometry of the densest packings in a uniform discrete set-up. We establish density bounds, optimal configurations reaching them in all cases, and introduce a probabilistic cel...

We construct a dense packing of regular tetrahedra, with packing density D > .7786157.

The dense packing of particulate assemblies is an enduring theoretical and practical problem. In spite of its apparent simplicity, this problem still remains challenging. Recent progress in the performance of computers boosted the development and realization of a number of effective packing algorithms. Yet, because of the complexity of the problem, many of existing methods have difficulties in ...

The notion of a completely saturated packing [1] is a sharper version of maximum density, and the analogous notion of a completely reduced covering is a sharper version of minimum density. We define two related notions: uniformly recurrent dense packings and diffusively dominant packings. Every compact domain in Euclidean space has a uniformly recurrent dense packing. If the domain self-nests, ...