Emeric Gioan

Combinatorial geometries

This special volume is dedicated to the memory of late Michel Las Vergnas who passed away on January 19, 2013 at age 72. Michel Las Vergnaswas a combinatorialmathematicianwith artistic sense and creativity. He is also well known as a cofounder of the theory of oriented matroids which acquired the firm acceptance as fundamental concept in mathematics. Michel was one of the founding editors of European Journal of Combinatorics. On personal side,Michel had broad and deep interest inmusic, arts and culture.Michel was a warm-hearted individual caring for his family and friends with kind and genuine smiles. Michel was the first doctoral student of another great combinatorialist Claude Berge. They were two of the leaders of the Paris school of combinatorics (L’Equipe Combinatoire, Université Paris VI). They influenced greatly the advances of combinatorial mathematics in the world and guided many excellent graduate students. Both Claude and Michel had unusual talents to formulate beautiful conjectures. Throughout his whole career as Directeur de Recherche au CNRS and professor of Université Paris VI (Pierre et Marie Curie), Michel’s devotion to combinatorial mathematics and to the guidance of doctoral students was extraordinary. The late Yahya Ould Hamidoune and the first editor Raul were the first doctoral students ofMichel, graduated in 1978 and 1979, respectively. The third editor Emeric was the last student who completed his study in 2002. Michel had supervised 15 doctoral students. He won the silver medal of the CNRS in 1985. On his pioneering research front, Michel gave many equivalent axiomatizations of oriented matroids in an extensive manuscript,1 which unfortunately has never been fully published. Michel’s theorem on single-element extensions2 of oriented matroids turned out to be crucial in constructing fascinating examples and in resolving degeneracy in the abstract combinatorial setting of optimization. Michel had strong interest in a wide range of combinatorics beyond the oriented

The active bijection in graphs, hyperplane arrangements, and oriented matroids, 1: The fully optimal basis of a bounded region

The present paper is the first in a series of four dealing with a mapping, introduced by the present authors, from orientations to spanning trees in graphs, from regions to simplices in real hyperplane arrangements, from reorientations to bases in oriented matroids (in order of increasing generality). This mapping is actually defined for ordered oriented matroids. We call it the active orientation-to-basis mapping, in reference to an extensive use of activities, a notion depending on a linear ordering, first introduced by W.T. Tutte for spanning trees in graphs. The active mapping, which preserves activities, can be considered as a bijective generalization of a polynomial identity relating two expressions-one in terms of activities of reorientations, and the other in terms of activities of bases-of the Tutte polynomial of a graph, a hyperplane arrangement or an oriented matroid. Specializations include bijective versions of well-known enumerative results related to the counting of acyclic orientations in graphs or of regions in hyperplane arrangements. Other interesting features of the active mapping are links established between linear programming and the Tutte polynomial. We consider here the bounded case of the active mapping, where bounded corresponds to bipolar orientations in the case of graphs, and refers to bounded regions in the case of real hyperplane arrangements, or of oriented matroids. In terms of activities, this is the uniactive internal case. We introduce fully optimal bases, simply defined in terms of signs, strengthening optimal bases of linear programming. An optimal basis is associated with one flat with a maximality property, whereas a fully optimal basis is equivalent to a complete flag of flats, each with a maximality property. The main results of the paper are that a bounded region has a unique fully optimal basis, and that, up to negating all signs, fully optimal bases provide a bijection between bounded regions and uniactive internal bases. In the bounded case, up to negating all signs, the active mapping is a bijection.

Complete graph drawings up to triangle mutations

The logical structure we introduce here to describe a (topological) graph drawing, called subsketch, is intermediate between the map (determining the drawing when it is planar), and the sketch introduced by Courcelle (determining the drawing in general but assuming we know the order of the crossings on each edge). For a complete graph drawing, the subsketch is determined, through first order logic formulas, by the size, a corner of the drawing and the crossings of the edges. n nWe prove, constructively, that two complete graph drawings have the same subsketch if and only if they can be transformed into each other by a sequence of triangle mutations – or triangle switches. This construction generalizes Ringels theorem on uniform pseudoline arrangements. Moreover, it applies to plane projections of spatial graphs encoded by rank 4 uniform oriented matroids.

Mapping Computation with No Memory

We investigate the computation of mappings from a set S n to itself with in situ programs , that is using no extra variables than the input, and performing modifications of one component at a time. We consider several types of mappings and obtain effective computation and decomposition methods, together with upper bounds on the program length (number of assignments). Our technique is combinatorial and algebraic (graph coloration, partition ordering, modular arithmetics). n nFor general mappings, we build a program with maximal length 5n *** 4, or 2n *** 1 for bijective mappings. The length is reducible to 4n *** 3 when |S | is a power of 2. This is the main combinatorial result of the paper, which can be stated equivalently in terms of multistage interconnection networks as: any mapping of {0,1} n can be performed by a routing in a double n -dimensional Benes network. Moreover, the maximal length is 2n *** 1 for linear mappings when S is any field, or a quotient of an Euclidean domain (e.g. ***/s ***). In this case the assignments are also linear, thereby particularly efficient from the algorithmic viewpoint. n nThe in situ trait of the programs constructed here applies to optimization of program and chip design with respect to the number of variables, since no extra writing memory is used. In a non formal way, our approach is to perform an arbitrary transformation of objects by successive elementary local transformations inside these objects only with respect to their successive states.

In Situ Design of Register Operations

We present methods to design programs or electronic circuits, for performing any operation on k registers of any sizes in a processor, in such a way that one uses no other working memory (such as other registers or external memories). In this way, any operation is performed with at most 4k - 3 assignments of these registers, or 2k - 1 when the operation is linear or bijective.