Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, II: The mixed Dirichlet-Neumann Problem

In this paper we continue the study started in part I (posted). We consider a planar, bounded, $m$-connected region $\Omega$, and let \bord\Omega be its boundary. Let $\mathcal{T}$ be a cellular decomposition of \Omega\cup\bord\Omega, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair $(S,f)$ where $S$ is a special type of a (possibly immersed) genus $(m-1)$ singular flat surface, tiled by rectangles and $f$ is an energy preserving mapping from ${\mathcal T}^{(1)}$ onto $S$. In part I the solution of a Dirichlet problem defined on ${\mathcal T}^{(0)}$ was utilized, in this paper we employ the solution of a mixed Dirichlet-Neumann problem.Comment: 26 pages, 16 figures (color

Perfect graphs of fixed density: counting and homogenous sets

For c in [0,1] let P_n(c) denote the set of n-vertex perfect graphs with density c and C_n(c) the set of n-vertex graphs without induced C_5 and with density c. We show that log|P_n(c)|/binom{n}{2}=log|C_n(c)|/binom{n}{2}=h(c)+o(1) with h(c)=1/2 if 1/4<c<3/4 and h(c)=H(|2c-1|)/2 otherwise, where H is the binary entropy function. Further, we use this result to deduce that almost all graphs in C_n(c) have homogenous sets of linear size. This answers a question raised by Loebl, Reed, Scott, Thomason, and Thomass\'e [Almost all H-free graphs have the Erd\H{o}s-Hajnal property] in the case of forbidden induced C_5.Comment: 19 page

Minimal surfaces in Riemannian Fibrations

In this dissertation, minimal and constant mean curvature surface theory in 3-dimensional Riemannian fibrations are studied. In the main part of the thesis, new complete, embedded minimal surfaces in the 3-sphere are constructed by solving a Plateau problem with respect to a suitable Jordan curve consisting entirely of horizontal geodesic arcs and extending this solution by means of Schwarz reflection. Additionally, an elementary proof for the vertical half-space theorem in Heisenberg space is given by finding a subsolution of the minimal surface equation. Finally, projections of constant mean curvature multigraphs are characterized: they are locally contained to one side of complete curves with constant geodesic curvature

A 7/9 - Approximation Algorithm for the Maximum Traveling Salesman Problem

We give a 7/9 - Approximation Algorithm for the Maximum Traveling Salesman Problem.Comment: 6 figure