Vertex covers by monochromatic pieces - A survey of results and problems

This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects. It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014.Comment: Discrete Mathematics, 201

Trivalent expanders and hyperbolic surfaces

We introduce a family of trivalent expanders which tessellate compact hyperbolic surfaces with large isometry groups. We compare this family with Platonic graphs and modifications of them and prove topological and spectral properties of these families

Termination Detection of Local Computations

Contrary to the sequential world, the processes involved in a distributed system do not necessarily know when a computation is globally finished. This paper investigates the problem of the detection of the termination of local computations. We define four types of termination detection: no detection, detection of the local termination, detection by a distributed observer, detection of the global termination. We give a complete characterisation (except in the local termination detection case where a partial one is given) for each of this termination detection and show that they define a strict hierarchy. These results emphasise the difference between computability of a distributed task and termination detection. Furthermore, these characterisations encompass all standard criteria that are usually formulated : topological restriction (tree, rings, or triangu- lated networks ...), topological knowledge (size, diameter ...), and local knowledge to distinguish nodes (identities, sense of direction). These results are now presented as corollaries of generalising theorems. As a very special and important case, the techniques are also applied to the election problem. Though given in the model of local computations, these results can give qualitative insight for similar results in other standard models. The necessary conditions involve graphs covering and quasi-covering; the sufficient conditions (constructive local computations) are based upon an enumeration algorithm of Mazurkiewicz and a stable properties detection algorithm of Szymanski, Shi and Prywes

High-dimensional fractionalization and spinon deconfinement in pyrochlore antiferromagnets

The ground states of Klein type spin models on the pyrochlore and checkerboard lattice are spanned by the set of singlet dimer coverings, and thus possess an extensive ground--state degeneracy. Among the many exotic consequences is the presence of deconfined fractional excitations (spinons) which propagate through the entire system. While a realistic electronic model on the pyrochlore lattice is close to the Klein point, this point is in fact inherently unstable because any perturbation $\epsilon$ restores spinon confinement at $T = 0$. We demonstrate that deconfinement is recovered in the finite--temperature region $\epsilon \ll T \ll J$, where the deconfined phase can be characterized as a dilute Coulomb gas of thermally excited spinons. We investigate the zero--temperature phase diagram away from the Klein point by means of a variational approach based on the singlet dimer coverings of the pyrochlore lattices and taking into account their non--orthogonality. We find that in these systems, nearest neighbor exchange interactions do not lead to Rokhsar-Kivelson type processes.Comment: 19 page