Even maps, the Colin de~Verdi\`ere number and representations of graphs

Van der Holst and Pendavingh introduced a graph parameter $\sigma$, which coincides with the more famous Colin de Verdi\`{e}re graph parameter $\mu$ for small values. However, the definition of $\sigma$ is much more geometric/topological directly reflecting embeddability properties of the graph. They proved $\mu(G) \leq \sigma(G) + 2$ and conjectured $\mu(G) \leq \sigma(G)$ for any graph $G$. We confirm this conjecture. As far as we know, this is the first topological upper bound on $\mu(G)$ which is, in general, tight. Equality between $\mu$ and $\sigma$ does not hold in general as van der Holst and Pendavingh showed that there is a graph $G$ with $\mu(G) \leq 18$ and $\sigma(G)\geq 20$. We show that the gap appears on much smaller values, namely, we exhibit a graph $H$ for which $\mu(H)\leq 7$ and $\sigma(H)\geq 8$. We also prove that, in general, the gap can be large: The incidence graphs $H_q$ of finite projective planes of order $q$ satisfy $\mu(H_q) \in O(q^{3/2})$ and $\sigma(H_q) \geq q^2$.Comment: 28 pages, 4 figures. In v2 we slightly changed one of the core definitions (previously "extended representation" now "semivalid representation"). We also use it to introduce a new graph parameter, denoted eta, which did not appear in v1. It allows us to establish an extended version of the main result showing that mu(G) is at most eta(G) which is at most sigma(G) for every graph

Kommutative Algebra und algebraische Geometrie

[no abstract available

Edge-unfolding almost-flat convex polyhedral terrains

Thesis (M. Eng.)--Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2013.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 97-98).In this thesis we consider the centuries-old question of edge-unfolding convex polyhedra, focusing specifically on edge-unfoldability of convex polyhedral terrain which are "almost at" in that they have very small height. We demonstrate how to determine whether cut-trees of such almost-at terrains unfold and prove that, in this context, any partial cut-tree which unfolds without overlap and "opens" at a root edge can be locally extended by a neighboring edge of this root edge. We show that, for certain (but not all) planar graphs G, there are cut-trees which unfold for all almost-at terrains whose planar projection is G. We also demonstrate a non-cut-tree-based method of unfolding which relies on "slice" operations to build an unfolding of a complicated terrain from a known unfolding of a simpler terrain. Finally, we describe several heuristics for generating cut-forests and provide some computational results of such heuristics on unfolding almost-at convex polyhedral terrains.by Yanping Chen.M.Eng

Group field theory renormalization - the 3d case: power counting of divergences

We take the first steps in a systematic study of Group Field Theory renormalization, focusing on the Boulatov model for 3D quantum gravity. We define an algorithm for constructing the 2D triangulations that characterize the boundary of the 3D bubbles, where divergences are located, of an arbitrary 3D GFT Feynman diagram. We then identify a special class of graphs for which a complete contraction procedure is possible, and prove, for these, a complete power counting. These results represent important progress towards understanding the origin of the continuum and manifold-like appearance of quantum spacetime at low energies, and of its topology, in a GFT framework