We consider the problem of determining finite integer sets which are knapsack-solvable in linear time (that is, it is possible to determine in linear time, for any integer b, whether b can be expressed as a sum of distinct elements of that set) and where the largest element is as small as possible. We study the condition that the k-subsets (for fixed k) when lexicographically ordered have incre...

A finite group is said to have "perfect order classes" if the number of elements any given either zero or a divisor group. The purpose this note describe explicitly Hamiltonian groups with perfect classes. We show that has classes if, and only it isomorphic direct product quaternion $8$, non-trivial cyclic $3$-group at most $2$. Theorem. $Q\times C_{3^k}$ C_{2}\times C_{3^k}$, for some positive...

A total perfect code in a graph is a subset of the graph’s vertices with the property that each vertex in the graph is adjacent to exactly one vertex in the subset. We prove that the tensor product of any number of simple graphs has a total perfect code if and only if each factor has a total perfect code.

A perfect r-code in a graph is a subset of the graph’s vertices with the property that each vertex in the graph is within distance r of exactly one vertex in the subset. We prove that the n-fold strong product of simple graphs has a perfect r-code if and only if each factor has a perfect r-code.

Faudree, R.J., R.J. Gould and L.M. Lesniak, Neighborhood conditions and edge-disjoint perfect matchings, Discrete Mathematics 91 (1991) 33-43. A graph G satisfies the neighborhood condition ANC(G) 2 m if, for all pairs of vertices of G, the union of their neighborhoods has at least m vertices. For a fixed positive integer k, let G be a graph of even order n which satisfies the following conditi...

Given a multiset of n positive integers, the NP-complete problem of number partitioning is to assign each integer to one of k subsets, such that the largest sum of the integers assigned to any subset is minimized. Last year, three different papers on optimally solving this problem appeared in the literature, two from the first two authors, and one from the third author. We resolve here competin...

For a not empty perfect subset of the unit circle C there is a perfect subset of C measure zero which being rotated to every position intersects the first set on a nonempty perfect set. This result may be stated in terms of set of distances between pairs of points from these two sets. A generalization of this result to a product of tori is suggested. 2000 Mathematics Subject Classification: 26-...

This paper studies the use of sensitivities to quantify the extent to which individual airline departments (stakeholders) contribute to the incident of runway overrun. For that purpose, we present a model of the incident runway overrun. The incident model is based on the dynamics of aircraft and describes the functional relationship between contributing factors leading to the incident. The inci...