Abstract List coloring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening this problem, where instead one list‐coloring, we seek many parallel. Our explorations have uncovered potentially rich seam interesting problems spanning chromatic Given ‐list‐assignment , which assignment list colors to each vertex existence pairwise‐disjoint proper colo...

The two-dimensional discrete bin-packing problem (2BP ) consists in minimizing the number of identical rectangles used to pack a set of smaller rectangles. This problem is NP-complete. It occurs in industry if pieces of steel, wood, or paper have to be cut from larger rectangles. It belongs to the family of cutting and packing (C & P) problems, more precisely Two-Dimensional Single Bin Size Bin...

In this paper we consider a more complicated real-world problem originating in the steel industry. The bins are inhomogeneous sheets with impurities. We assume that each impure area is rectangle. For each bin we are given a set of impurities, size, and location of each impurity into the bin. As a consequence now the bins are not identical anymore and the number of bins is finite. Moreover, we i...

Let H be any graph. We determine (up to an additive constant) the minimum degree of a graph G which ensures that G has a perfect H-packing. More precisely, let δ(H, n) denote the smallest integer k such that every graph G whose order n is divisible by |H| and with δ(G) ≥ k contains a perfect H-packing. We show that

We study the two-dimensional bin packing problem with and without rotations. Here we are given a set of two-dimensional rectangular items I and the goal is to pack these into a minimum number of unit square bins. We consider the orthogonal packing case where the edges of the items must be aligned parallel to the edges of the bin. Our main result is a 1.405approximation for two-dimensional bin p...

In order to verify and test the performance of new packing algorithms relative to existing algorithms, test problems are needed. The scope of published test instances for 2D rectangular and irregular packing has been fairly limited in terms of object size, number and dimension of items. A systematic investigation into the performance of new algorithms requires the use of a range of test problem...

Let X = {X(t), t ∈ R+} be an operator stable Lévy process in R with exponent B, where B is an invertible linear operator on R. We determine the Hausdorff dimension and the packing dimension of the range X([0, 1]) in terms of the real parts of the eigenvalues of B. Running Title Dimension of Operator Stable Lévy Processes

This paper deals with the densest packing of equal circles in a square problem. Sharp bounds for the density of optimal circle packings have given. Several known optimal and approximate circle packings contain optimal substructures. Based on this observation it is sometimes easy to determine the minimal polynomials of the arrangements.

We have collected definitions and basic results for the (centered ball) density in metric space with respect to an arbitrary Hausdorff function. We have kept the definitions general: we do not assume the Hausdorff functions are continuous or blanketed, and we do not assume the metric space is a subset of Euclidean space. We discuss the covering measure (= centered Hausdorff measure) and packing...