Hung-Lin Fu

Connectivity of cages

A (k; g)-graph is a k-regular graph with girth g. Let f(k; g) be the smallest integer v such there exists a (k; g)-graph with v vertices. A (k; g)-cage is a (k; g)-graph with f(k; g) vertices. In this paper we prove that the cages are monotonic in that f(k; g1) < f(k; g2) for all k ≥ 3 and 3 ≥ g1 < g2. We use this to prove that (k; g)-cages are 2-connected, and if k = 3 then their connectivity is k.

Nonadaptive algorithms for threshold group testing

Threshold group testing first proposed by Damaschke is a generalization of classic group testing. Specifically, a group test is positive (negative) if it contains at least u (at most l) positives, and if the number of positives is between l and u, the test outcome is arbitrary. Although sequential group testing algorithms have been proposed, it is unknown whether an efficient nonadaptive algorithm exists. In this paper, we give an affirmative answer to this problem by providing efficient nonadaptive algorithms for the threshold model. The key observation is that disjunct matrices, a standard tool for group testing designs, also work in this threshold model. This paper improves and extends previous results in three ways: 1. The algorithms we propose work in one stage, which saves time for testing. 2. The test complexity is lower than previous results, at least for the number of elements which need to be tested is sufficiently large. 3. A limited number of erroneous test outcomes are allowed.

An upper bound of the number of tests in pooling designs for the error-tolerant complex model

Recently pooling designs have been used in screening experiments in molecular biology. In some applications, the property to be screened is defined on subsets of items, instead of on individual items. Such a model is usually referred to as the complex model. In this paper we give an upper bound of the number of tests required in a pooling design for the complex model (with given design parameters) where experimental errors are allowed.

On prime labellings

Let G = (V, E) be a graph. A bijection f: V ?{;1,2,?,|V|} is called a prime labelling if for each e = {u, v} in E, we have GCD(f(u),f(?)) = 1. A graph admits a prime labelling is called a prime graph. Around ten years ago, Roger Entriger conjectured that every tree is prime. So far, this conjecture is still unsolved. In this paper, we show that the conjecture is true for trees of order up to 15, and also show that a few other classes of graphs are prime.

Optimal Conflict-Avoiding Codes of Even Length and Weight 3

Direct constructions for optimal conflict-avoiding codes of length

Linear k -arboricities on trees

n \equiv 4 \pmod{8}

Optimal conflict-avoiding codes of odd length and weight three

and weight 3 are provided by bringing in a new concept called an extended odd sequence. Constructions for those odd sequences are also given in this paper. As a consequence, with previously known results, the spectrum of the size of optimal conflict-avoiding codes of even length and weight 3 is completely settled.

Reconstruction of hidden graphs and threshold group testing

For a xed positive integer k, the linear k-arboricity lak(G) of a graph G is the minimum number ‘ such that the edge set E(G) can be partitioned into ‘ disjoint sets and that each induces a subgraph whose components are paths of lengths at most k. This paper studies linear k-arboricity from an algorithmic point of view. In particular, we present a linear-time algorithm to determine whether a tree T has lak(T)6m. ? 2000 Elsevier Science B.V. All rights reserved.

Packing complete multipartite graphs with 4-cycles

A conflict-avoiding code (CAC)