New construction for transversal design.

The study of gene functions requires the development of a DNA library of high quality through much of testing and screening. Pooling design is a mathematical tool to reduce the number of tests for DNA library screening. The transversal design is a special type of pooling design, which is good in implementation. In this paper, we present a new construction for transversal designs. We will also extend our construction to the error-tolerant case.

Error-tolerant pooling designs with inhibitors.

Pooling designs are used in clone library screening to efficiently distinguish positive clones from negative clones. Mathematically, a pooling design is just a nonadaptive group testing scheme which has been extensively studied in the literature. In some applications, there is a third category of clones called "inhibitors" whose effect is to neutralize positives. Specifically, the presence of an inhibitor in a pool dictates a negative outcome even though positives are present. Sequential group testing schemes, which can be modified to three-stage schemes, have been proposed for the inhibitor model, but it is unknown whether a pooling design (a one-stage scheme) exists. Another open question raised in the literature is whether the inhibitor model can treat unreliable pool outcomes. In this paper, we answer both open problems by giving a pooling design, as well as a two-stage scheme, for the inhibitor model with unreliable outcomes. The number of pools required by our schemes are quite comparable to the three-stage scheme.

The Steiner ratio conjecture of Gilbert and Pollak is true.

Let P be a set of n points on the euclidean plane. Let Ls(P) and Lm(P) denote the lengths of the Steiner minimum tree and the minimum spanning tree on P, respectively. In 1968, Gilbert and Pollak conjectured that for any P, Ls(P) >/= (radical3/2)Lm(P). We provide an abridged proof for their conjecture in this paper.