The study of gene functions requires a DNA library of high quality, such a library is obtained from a large mount of testing and screening. Pooling design is a very helpful tool for reducing the number of tests for DNA library screening. In this paper, we present new one- and two-stage pooling designs, together with new probabilistic pooling designs. The approach in this paper works for both error-free and error-tolerance scenarios.

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An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1, . . . , m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every connected graph, but K 2 , is antimagic. In 2004, N. Alon et al showed that this conjecture is true for n-vertex graphs with minimum degree Ω(log n). They also proved that complete partite graphs (other than K 2 ) and n-vertex graphs with maximum degree at least n − 2 are antimagic. Recently, Wang showed that the toroidal grids (the Cartesian products of two or more cycles) are antimagic. Two open problems left in Wang's paper are about the antimagicness of lattice grid graphs and prism graphs, which are the Cartesian products of two paths, and of a cycle and a path, respectively. In this article, we prove that these two classes of graphs are antimagic, by constructing such antimagic labelings.

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Abstract. The firefighter problem is the following discrete-time game on a graph. Initially, a fire starts at a vertex of the graph. In each round, a firefighter protects one vertex not yet on fire, and then the fire spreads to all unprotected neighbors of the vertices on fire. The objective of the firefighter is to save as many vertices as possible. The surviving rate of a graph is the average percentage of vertices that can be saved when a fire starts randomly at one vertex of the graph, which measures the defense ability of a graph as a whole. In this paper, we study the surviving rates of graphs with bounded treewidth. We prove that the surviving rate of every n-vertex outerplanar graph is at least 1 − Θ( log n n ), which is asymptotically tight. We also prove that if k firefighters are available in each round, then the surviving rate of an n-vertex graph with treewidth at most k is 1. Introduction. The firefighter problem is a discrete-time game on graphs introduced by Hartnell [8] at a conference in 1995, who attempted to model firefighting or virus control on a network. The game goes as follows. A fire breaks out at a vertex of a graph G = (V, E), and then the fire and a firefighter make alternate moves on the graph. In each round, the firefighter protects at most one vertex not yet on fire, and the fire then spreads from all burning vertices (i.e., vertices on fire) to all their unprotected neighbors. Once a vertex is burning or protected, it remains so during the whole process. The process ends when the fire can no longer spread. All vertices that are not burning are saved. The main objective of the firefighter is to save as many vertices as possible.Various aspects of the firefighter problem have been studied in the literature. Finbow et al. [5] showed that it is NP-hard for the firefighter to save the maximum number of vertices, even for trees of maximum degree three. Hartnell and Li [9] proved that a simple greedy method for trees is a 0.5-approximation algorithm, and MacGillivray and Wang [11] gave a 0-1 integer programming formulation of the prob-