Ricardo Strausz

Separoids, Their Categories and a Hadwiger-Type Theorem for Transversals

In this paper we study the topology of transversals to a family of convex sets as a subset of a Grassmanian manifold. This topology seems to be ruled by a combinatorial structure which we call a separoid. With these combinatorial objects and the topological notion of virtual transversal we prove a Borsuk—Ulam-type theorem which has as a corollary a generalization of Hadwiger’s theorem.

A Generalisation of Tverberg’s Theorem

We will prove the following generalisation of Tverberg’s Theorem: given a set S⊂ℝd of (r+1)(k−1)(d+1)+1 points, there is a partition of S in k sets A1,A2,…,Ak such that for any C⊂S of at most r points, the convex hulls of A1\C,A2\C,…,Ak\C are intersecting. This was conjectured first by Natalia García-Colín (Ph.D. thesis, University College of London, 2007).

On the pseudoachromatic index of the complete graph

Let q = 2β be, for some β∈ℕ, and let n = q2 + q+ 1. By exhibiting a complete coloring of the edges of Kn, we show that the pseudoachromatic number ψ(Gn) of the complete line graph Gn = L(Kn)—or the pseudoachromatic index of Kn, if you will—is at least q3 + q. This bound improves the implicit bound of Jamison [Discrete Math 74 (1989), 99–115] which is given in terms of the achromatic number: ψ(Gn)≥α(Gn)≥q3 + 1. We also calculate, precisely, the pseudoachromatic number when q+ 1 extra points are added: Copyright

Applying DNA computing to diagnose-and-interfere hepatic fibrosis

In the last few years, advances in molecular computing have been developed fast enough to offer several paradigms of diagnosis and therapeutic applications. Herein we describe how molecular computing can be applied to detect activated hepatic stellate cells (HSC) -which are well known as the main responsible of hepatic fibrogenesis- and, as a response to positive diagnosis, express a gene (PPARγ1) which have been proved to reverse the myofibroblast-like phenotype of the HSC in vivo. We also propose a transport method to deliver the DNA computer into the HSC; namely, vitamin A-coupled liposomes.

Counting polytopes via the Radon complex

A convex polytope is the convex hull of a finite set of points. We introduce the Radon complex of a polytope--a subcomplex of an appropriate hypercube which encodes all Radon partitions of the polytopes vertex set. By proving that such a complex, when the vertices of the polytope are in general position, is homeomorphic to a sphere, we find an explicit formula to count the number of d-dimensional polytope types with d + 3 vertices in general position.

Cubic time recognition of cocircuit graphs of uniform oriented matroids

We present an algorithm which takes a graph as input and decides in cubic time whether the graph is the cocircuit graph of a uniform oriented matroid. In the affirmative case the algorithm returns the set of signed cocircuits of the oriented matroid. This improves an algorithm proposed by Babson, Finschi and Fukuda. Moreover we strengthen a result of Montellano-Ballesteros and Strausz characterizing cocircuit graphs of uniform oriented matroids in terms of crabbed connectivity.

Carathéodory-type Theorems à la Bárány

We generalise the famous Helly–Lovász theorem leading to a generalisation of the Bárány–Carathéodory theorem for oriented matroids in dimension ≤3. We also provide a non-metric proof of the latter colourful theorem for arbitrary dimensions and explore some generalisations in dimension 2.

On panchromatic patterns

Since the classic book of Berge (1985) it is well known that every digraph contains a kernel by paths. This was generalised by Sands etźal. (1982) who proved that every edge two-coloured digraph has a kernel by monochromatic paths. More generally, given D and H two digraphs, D is H -coloured iff the arcs of D are coloured with the vertices of H . Furthermore, an H -walk in D is a sequence of arcs forming a walk in D whose colours are a walk in H . With this notion of H -walks, we can define H -independence, which is the absence of such a walk pairwise, and H -absorbance, which is the existence of such a walk towards the absorbent set. Thus, an H -kernel is a subset of vertices which is both H -independent and H -absorbent. The aim of this paper is to characterise those H , which we call panchromatic patterns, for which all D and all H -colourings of D admits an H -kernel. This solves a problem of Arpin and Linek from 2007 (Arpin and Linek, 2007).

Bounding the pseudoachromatic index of the complete graph via projective planes

Abstract Let q = 2 β and n = q 2 + q + 1 . Further, let G = L ( K n ) be the complete line graph and ψ ( G ) its pseudoachromatic number. By exhibiting an explicit colouring of E ( K n ) , we show that ψ ( G ) ⩾ q 3 + q . This result improves the bound ψ ( G ) ⩾ q 3 + 1 due to Jamison (1989) [Jamison, R.E.; On the edge achromatic numbers of complete graphs, Discrete. Math. 74 (1989) 99–115].

On the Pseudoachromatic Index of the Complete Graph III

An edge colouring of a graph G is complete if for any distinct colours