Clemens Huemer

On the number of plane graphs

We investigate the number of plane geometric, i.e., straight-line, graphs, a set <i>S</i> of <i>n</i> points in the plane admits. We show that the number of plane graphs and connected plane graphs as well as the number of cycle-free plane graphs is minimized when <i>S</i> is in convex position. Moreover, these results hold for all these graphs with an arbitrary but fixed number of edges. Consequently, we provide simple proofs that the number of spanning trees, cycle-free graphs (forests), perfect matchings, and spanning paths is also minimized for point sets in convex position.In addition we construct a new extremal configuration, the so-called double zig-zag chain. Most noteworthy this example bears Θ*(√72^<i>n</i>) = Θ*(8.4853^<i>n</i>) triangulations and Θ*(41.1889^<i>n</i>) plane graphs (omitting polynomial factors in both cases), improving the previously known best maximizing examples.

On k-gons and k-holes in point sets

We consider a variation of the classical Erd?s-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.

Empty monochromatic triangles

We consider a variation of a problem stated by Erdos and Guy in 1973 about the number of convex k-gons determined by any set S of n points in the plane. In our setting the points of S are colored and we say that a spanned polygon is monochromatic if all its points are colored with the same color. As a main result we show that any bi-colored set of n points in R^2 in general position determines a super-linear number of empty monochromatic triangles, namely @W(n^5^/^4).

Large Bichromatic Point Sets Admit Empty Monochromatic 4-Gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).

Triangulations without pointed spanning trees

Problem 50 in the Open Problems Project of the computational geometry community asks whether any triangulation on a point set in the plane contains a pointed spanning tree as a subgraph. We provide a counterexample. As a consequence we show that there exist triangulations which require a linear number of edge flips to become Hamiltonian.

4-Holes in point sets

We consider a variant of a question of Erd?s on the number of empty k-gons (k-holes) in a set of n points in the plane, where we allow the k-gons to be non-convex. We show bounds and structural results on maximizing and minimizing the number of general 4-holes, and maximizing the number of non-convex 4-holes. In particular, we show that for n ? 9 , the maximum number of general 4-holes is ( n 4 ) ; the minimum number of general 4-holes is at least 5 2 n 2 - ? ( n ) ; and the maximum number of non-convex 4-holes is at least 1 2 n 3 - ? ( n 2 log n ) and at most 1 2 n 3 - ? ( n 2 ) .

On the Fiedler value of large planar graphs

Abstract 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 ( 1 n 2 ) ⩽ λ 2 max ⩽ 2 + O ( 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.

4-Labelings and grid embeddings of plane quadrangulations

Finding aesthetic drawings of planar graphs is a main issue in graph drawing. Of special interest are rectangle of influence drawings.The graphs considered here are quadrangulations, that is, planar graphs all whose faces have degree four.We show that each quadrangulation on n vertices has a closed rectangle of influence drawing on the (n - 2) × (n - 2) grid. Biedl, Bretscher and Meijer [2] proved that every planar graph on n vertices without separating triangle has a closed rectangle of influence drawing on the (n - 1) × (n - 1) grid.Our method, which is completely different from that of [2], is in analogy to Schnyders algorithm for embedding triangulations on an integer grid [9] and gives a simple algorithm.

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 ? ( 12.5 2 N ) non-crossing spanning trees and ? ( 13.6 1 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 Toth (2013). 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 ( 22.1 2 N ) for the number of non-crossing spanning trees of the double chain is also obtained.

Gray codes for non-crossing partitions and dissections of a convex polygon

In this paper we develop Gray codes for two families of geometric objects: non-crossing partitions and dissections of a convex polygon by means of non-crossing diagonals.