arXiv subjects

Clemens Huemer

Publications and source records attributed to Clemens Huemer.

26 records · Page 2Linked to original sources

arXiv2014

On $k$-Gons and $k$-Holes in Point Sets

We consider a variation of the classical Erdős-Szekeres problems on the existence and number of convex $k$-gons and $k$-holes (empty $k$-gons) in a set of $n$ points in the plane. Allowing the $k$-gons to be non-convex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any $k$ and sufficiently large $n$, we give a quadratic lower bound for the number of $k$-holes, and show that this number is maximized by sets in convex position.

cs.DM↗

arXiv2013

Lower bounds on the maximum number of non-crossing acyclic graphs

This paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of non-crossing spanning trees and forests. We show that the so-called double chain point configuration of N points has Omega(12.52^N) non-crossing spanning trees and Omega(13.61^N) non-crossing forests. This improves the previous lower bounds on the maximum number of non-crossing spanning trees and of non-crossing forests among all sets of N points in general position given by Dumitrescu, Schulz, Sheffer and Tóth. Our analysis relies on the tools of analytic combinatorics, which enable us to count certain families of forests on points in convex position, and to estimate their average number of components. A new upper bound of O(22.12^N) for the number of non-crossing spanning trees of the double chain is also obtained.

math.CO↗

arXiv2013

On the Fiedler value of large planar graphs

The Fiedler value $λ_2$, also known as algebraic connectivity, is the second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler value among all planar graphs $G$ with $n$ vertices, denoted by $λ_{2\max}$, and we show the bounds $2+Θ(\frac{1}{n^2}) \leq λ_{2\max} \leq 2+O(\frac{1}{n})$. We also provide bounds on the maximum Fiedler value for the following classes of planar graphs: Bipartite planar graphs, bipartite planar graphs with minimum vertex degree~3, and outerplanar graphs. Furthermore, we derive almost tight bounds on $λ_{2\max}$ for two more classes of graphs, those of bounded genus and $K_h$-minor-free graphs.

math.CO↗

arXiv2012

Empty Monochromatic Simplices

Let $S$ be a $k$-colored (finite) set of $n$ points in $\mathbb{R}^d$, $d\geq 3$, in general position, that is, no {$(d + 1)$} points of $S$ lie in a common $(d - 1)$}-dimensional hyperplane. We count the number of empty monochromatic $d$-simplices determined by $S$, that is, simplices which have only points from one color class of $S$ as vertices and no points of $S$ in their interior. For $3 \leq k \leq d$ we provide a lower bound of $Ω(n^{d-k+1+2^{-d}})$ and strengthen this to $Ω(n^{d-2/3})$ for $k=2$. On the way we provide various results on triangulations of point sets in $\mathbb{R}^d$. In particular, for any constant dimension $d\geq3$, we prove that every set of $n$ points ($n$ sufficiently large), in general position in $\mathbb{R}^d$, admits a triangulation with at least $dn+Ω(\log n)$ simplices.

math.CO↗

arXiv2009

Token Graphs

For a graph $G$ and integer $k\geq1$, we define the token graph $F_k(G)$ to be the graph with vertex set all $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is a pair of adjacent vertices in $G$. Thus vertices of $F_k(G)$ correspond to configurations of $k$ indistinguishable tokens placed at distinct vertices of $G$, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex. This paper introduces token graphs and studies some of their properties including: connectivity, diameter, cliques, chromatic number, Hamiltonian paths, and Cartesian products of token graphs.

math.CO↗

arXiv2009

Maximizing Maximal Angles for Plane Straight-Line Graphs

Let $G=(S, E)$ be a plane straight-line graph on a finite point set $S\subset\R^2$ in general position. The incident angles of a vertex $p \in S$ of $G$ are the angles between any two edges of $G$ that appear consecutively in the circular order of the edges incident to $p$. A plane straight-line graph is called $ϕ$-open if each vertex has an incident angle of size at least $ϕ$. In this paper we study the following type of question: What is the maximum angle $ϕ$ such that for any finite set $S\subset\R^2$ of points in general position we can find a graph from a certain class of graphs on $S$ that is $ϕ$-open? In particular, we consider the classes of triangulations, spanning trees, and paths on $S$ and give tight bounds in most cases.

cs.CG↗

arXiv2008

Binary Labelings for Plane Quadrangulations and their Relatives

Motivated by the bijection between Schnyder labelings of a plane triangulation and partitions of its inner edges into three trees, we look for binary labelings for quadrangulations (whose edges can be partitioned into two trees). Our labeling resembles many of the properties of Schnyder's one for triangulations: Apart from being in bijection with tree decompositions, paths in these trees allow to define the regions of a vertex such that counting faces in them yields an algorithm for embedding the quadrangulation, in this case on a~2-book. Furthermore, as Schnyder labelings have been extended to 3-connected plane graphs, we are able to extend our labeling from quadrangulations to a larger class of 2-connected bipartite graphs. Finally, we propose a binary labeling for Laman graphs.

math.CO↗

arXiv2008

Compatible Geometric Matchings

This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are \emph{compatible} if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings $M$ and $M'$ of the same set of $n$ points, for some $k\in\Oh{\log n}$, there is a sequence of perfect matchings $M=M_0,M_1,...,M_k=M'$, such that each $M_i$ is compatible with $M_{i+1}$. This improves the previous best bound of $k\leq n-2$. We then study the conjecture: \emph{every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching}. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with $n$ edges has an edge-disjoint compatible matching with approximately $4n/5$ edges.

math.CO↗