arXiv subjects

Aaron Büngener

Publications and source records attributed to Aaron Büngener.

6 recordsLinked to original sources

arXiv2026

Embedding Trees and Cactus Graphs with Integer Edge Lengths on a Compact Integer Grid

Every planar graph can be embedded with crossing-free, straight-line edges on an $O(n) \times O(n)$ integer grid, where $n$ is the number of vertices. It is not known, however, whether integer edge lengths can be guaranteed for all planar graphs (conjectured to be true by Harborth). Kleber conjectures something even stronger: Every planar graph admits a planar straight-line grid drawing where every edge has integer length. Positive results for Kleber's conjecture are known for several classes of planar graphs, however, all but one of the existing results are existential and do not provide bounds on the size of the integer grid. We address this gap for trees and cactus graphs. For trees, we present a linear-time algorithm that computes a crossing-free straight-line drawing with integer edge lengths on an $O(n \log n) \times O(n \log n)$ grid. We complement this result by giving an $Ω(n^2)$ lower bound. For cactus graphs, we present a linear-time algorithm that computes a crossing-free straight-line drawing with integer edge lengths on an $O(n^2 \log n) \times O(n^2 \log n)$ grid.

cs.CG↗

arXiv2026

A first view on the density of 5-planar graphs

A key concept for many graph layout algorithms is planarity, a graph property that allows to draw vertices and edges crossing-free in the plane. Important is the generalization to $k$-planar graphs, which can be drawn in the plane with at most $k > 0$ crossings per edge. One of the basic graph properties that have been explored for those graph classes is the maximum edge density, i.e., the maximum number of edges a $k$-planar graph on $n$ vertices may have. While there are numerous results for the classes of $1$- and $2$-planar graphs, there are few results for increasing $k=3$ or $4$ due to the complex graph structures. We make a first step towards even larger $k>4$ exploring the class of $5$-planar graphs. While our main tool is still a discharging technique, a better understanding of the structure of the denser parts leads to corresponding density bounds in a much simpler way. We first apply a simplified version of our technique to outer $5$-planar graphs and surprisingly observe that the structure of maximally dense (general) $5$-planar graphs differs from the known uniform structure of maximally dense $k$-planar graphs for smaller $k \in \{1,2,3,4\}$. As the central result of this paper, we then show that graphs that admit a simple 5-planar drawing have at most $7(n-2)$ edges, drastically improving the previous best bound of $\approx8.3n$. This even implies a small improvement of the leading constant in the Crossing Lemma $cr(G) \ge c \frac{m^3}{n^2}$ from $c=\frac{1}{27.48}$ to $c=\frac{1}{27.3}$. To demonstrate the potential of our new technique, we also apply it to 4-planar and 6-planar graphs.

cs.DM↗

arXiv2026

A Simplified Proof for the Edge-Density of 4-Planar Graphs

A graph on $n \ge 3$ vertices drawn in the plane such that each edge is crossed at most four times has at most $6(n-2)$ edges -- this result, proven by Ackerman, is outstanding in the literature of beyond-planar graphs with regard to its tightness and the structural complexity of the graph class. We provide a much shorter proof while at the same time relaxing the conditions on the graph and its embedding, i.e., allowing multi-edges and non-simple drawings.

math.CO↗

arXiv2024

Improving the Crossing Lemma by Characterizing Dense 2-Planar and 3-Planar Graphs

The classical Crossing Lemma by Ajtai et al.~and Leighton from 1982 gave an important lower bound of $c \frac{m^3}{n^2}$ for the number of crossings in any drawing of a given graph of $n$ vertices and $m$ edges. The original value was $c= 1/100$, which then has gradually been improved. Here, the bounds for the density of $k$-planar graphs played a central role. Our new insight is that for $k=2,3$ the $k$-planar graphs have substantially fewer edges if specific local configurations that occur in drawings of $k$-planar graphs of maximum density are forbidden. Therefore, we are able to derive better bounds for the crossing number $\text{cr}(G)$ of a given graph $G$. In particular, we achieve a bound of $\text{cr}(G) \ge \frac{37}{9}m-\frac{155}{9}(n-2)$ for the range of $5n < m \le 6n$, while our second bound $\text{cr}(G) \ge 5m - \frac{203}{9}(n-2)$ is even stronger for larger $m>6n$. For $m > 6.77n$, we finally apply the standard probabilistic proof from the BOOK and obtain an improved constant of $c>1/27.48$ in the Crossing Lemma. Note that the previous constant was $1/29$. Although this improvement is not too impressive, we consider our technique as an important new tool, which might be helpful in various other applications.

math.CO↗

arXiv2024

On $k$-planar Graphs without Short Cycles

We study the impact of forbidding short cycles to the edge density of $k$-planar graphs; a $k$-planar graph is one that can be drawn in the plane with at most $k$ crossings per edge. Specifically, we consider three settings, according to which the forbidden substructures are $3$-cycles, $4$-cycles or both of them (i.e., girth $\ge 5$). For all three settings and all $k\in\{1,2,3\}$, we present lower and upper bounds on the maximum number of edges in any $k$-planar graph on $n$ vertices. Our bounds are of the form $c\,n$, for some explicit constant $c$ that depends on $k$ and on the setting. For general $k \geq 4$ our bounds are of the form $c\sqrt{k}n$, for some explicit constant $c$. These results are obtained by leveraging different techniques, such as the discharging method, the recently introduced density formula for non-planar graphs, and new upper bounds for the crossing number of $2$-- and $3$-planar graphs in combination with corresponding lower bounds based on the Crossing Lemma.

math.CO↗

arXiv2024

Min-$k$-planar Drawings of Graphs

The study of nonplanar drawings of graphs with restricted crossing configurations is a well-established topic in graph drawing, often referred to as beyond-planar graph drawing. One of the most studied types of drawings in this area are the $k$-planar drawings $(k \geq 1)$, where each edge cannot cross more than $k$ times. We generalize $k$-planar drawings, by introducing the new family of min-$k$-planar drawings. In a min-$k$-planar drawing edges can cross an arbitrary number of times, but for any two crossing edges, one of the two must have no more than $k$ crossings. We prove a general upper bound on the number of edges of min-$k$-planar drawings, a finer upper bound for $k=3$, and tight upper bounds for $k=1,2$. Also, we study the inclusion relations between min-$k$-planar graphs (i.e., graphs admitting min-$k$-planar drawings) and $k$-planar graphs. In our setting we only allow simple drawings, that is, any two edges cross at most once, no two adjacent edges cross, and no three edges intersect at a common crossing point.

cs.CG↗