A first view on the density of 5-planar graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909044630880256 |
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| author | Büngener, Aaron Franz, Jakob Kaufmann, Michael Pfister, Maximilian |
| author_facet | Büngener, Aaron Franz, Jakob Kaufmann, Michael Pfister, Maximilian |
| contents | 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 $1 \leq k \leq 4$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_24364 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A first view on the density of 5-planar graphs Büngener, Aaron Franz, Jakob Kaufmann, Michael Pfister, Maximilian Discrete Mathematics Combinatorics 05C62, 05C10 G.2.2 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 $1 \leq k \leq 4$. 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. |
| title | A first view on the density of 5-planar graphs |
| topic | Discrete Mathematics Combinatorics 05C62, 05C10 G.2.2 |
| url | https://arxiv.org/abs/2505.24364 |