Group Testing with Pools of Fixed Size

In the classical combinatorial (adaptive) group testing problem, one is given two integers $d$ and $n$, where $0\le d\le n$, and a population of $n$ items, exactly $d$ of which are known to be defective. The question is to devise an optimal sequential algorithm that, at each step, tests a subset of the population and determines whether such subset is contaminated (i.e. contains defective items) or otherwise. The problem is solved only when the $d$ defective items are identified. The minimum number of steps that an optimal sequential algorithm takes in general (i.e. in the worst case) to solve the problem is denoted by $M(d, n)$. The computation of $M(d, n)$ appears to be very difficult and a general formula is known only for $d = 1$. We consider here a variant of the original problem, where the size of the subsets to be tested is restricted to be a fixed positive integer $k$. The corresponding minimum number of tests by a sequential optimal algorithm is denoted by $M^{\lbrack k\rbrack}(d, n)$. In this paper we start the investigation of the function $M^{\lbrack k\rbrack}(d, n)$

The list-chromatic index of K 6

We prove that the list-chromatic index and paintability index of K"6 is 5. That indeed @g"@?^'(K"6)=5 was a still open special case of the List Coloring Conjecture. Our proof demonstrates how colorability problems can numerically be approached by the use of computer algebra systems and the Combinatorial Nullstellensatz

Excessive [l,m]-factorizations

Given two positive integers l and m, with l 64m, an [l,m]-covering of a graph G is a set M of matchings of G whose union is the edge set of G and such that l 64;|M| 64m for every M. An [l,m]-covering M of G is an excessive [l,m]-factorization of G if the cardinality of M is as small as possible. The number of matchings in an excessive [l,m]-factorization of G (or 1e, if G does not admit an excessive [l,m]-factorization) is a graph parameter called the excessive [l,m]-index of G and denoted by \u3c7[l,m]\u2032(G). In this paper we study such parameter. Our main result is a general formula for the excessive [l,m]-index of a graph G in terms of other graph parameters. Furthermore, we give a polynomial time algorithm which computes \u3c7[l,m]\u2032(G) for any fixed constants l and m and outputs an excessive [l,m]-factorization of G, whenever the latter exists

A theorem in edge colouring

We prove the following theorem: if G is an edge-chromatic critical multigraph with more than 3 vertices, and if x, y are two adjacent vertices of G, the edge-chromatic number of G does not change if we add an extra edge joining x and y

The 1-Factorization Problem and some related Conjectures

E dalla crisalide sbucó una farfalla meravigliosa, tanto che tutti i fiori si aprirono ad essa. Out of the chrysalis there came a beautiful butterfly. It was so beautiful that all the flowers opened up to it. Daniela Rigato (1951-1996) The Classification Problem is the problem of determining whether or not a given graph is ∆-edge colourable, where ∆ is the maximum degree. This problem is known to be NP-hard, even when restricted to the class of cubic simple graphs. A theorem of Chetwynd and Hilton states that all regular graphs of order 2n and degree at least ( √ 7−1