Every $k$-Quasi-Planar Graph Has Linearly Many Edges

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1. Very close Same question Cites this paper Not relevant

   On the number of edges in quasiplanar graphs

   J. Pach, G. Tardos, Ji Zeng · 2026 · Not summarized yet

   On the number of edges in quasiplanar graphs presents a simplified proof that forbidding k pairwise crossing edges yields a linear edge bound.

   Abstract

   A $k$-quasiplanar graph is a graph drawn in the plane with no $k$ pairwise crossing edges. Pach, Shahrokhi, and Szegedy conjectured that every such graph on $n$ vertices has $O_k(n)$ edges. ChatGPT produced a proof of this conjecture. Here, we present a simplified, self-contained version of that proof.

2. Very close Another approach to the problem Cited by this paper Not relevant

   New bounds on the maximum number of edges in k-quasi-planar graphs

   Andrew Suk, Bartosz Walczak · 2013 · Computational geometry · cited 43 times · Not summarized yet

   New bounds studies the same edge-bound problem but gives near-linear estimates under bounded pairwise intersections, rather than a linear bound allowing repeated crossings.

   Abstract

   A topological graph is k-quasi-planar if it does not contain k pairwise crossing edges. A 20-year-old conjecture asserts that for every fixed k, the maximum number of edges in a k-quasi-planar graph on n vertices is O ( n ) . Fox and Pach showed that every k-quasi-planar graph with n vertices has at most n ( log ? n ) O ( log ? k ) edges. We improve this upper bound to 2 α ( n ) c n log ? n , where α ( n ) denotes the inverse Ackermann function and c depends only on k, for k-quasi-planar graphs in which any two edges intersect in a bounded number of points. We also show that every k-quasi-planar graph with n vertices in which any two edges have at most one point in common has at most O ( n l…

3. Very close* Same question Cited by this paper Not relevant

   Quasi-planar graphs have a linear number of edges

   P. Agarwal, B. Aronov, J. Pach et al. · 1995 · International Symposium Graph Drawing and Network Visualization · cited 150 times · Not summarized yet

   Quasi-planar graphs have a linear number of edges addresses whether forbidding pairwise-crossing edge sets forces a linear edge bound.

4. Close Broader context Not relevant

   The Exact Number of Pentagons in Book-Free Graphs

   Wen-Tao Hu · 2026 · Not summarized yet

   The Exact Number of Pentagons studies extremal counts for graphs avoiding book configurations, not crossing restrictions in graph drawings.

   Abstract

   Let $B_k$ denote the book consisting of $k$ triangles sharing a common edge. We prove that there exist absolute constants $\rho>0$ and $N$ such that, for every pair of integers $n,k$ with $n\ge N$ and $2\le k\le \rho n$, writing $n=5q+r$ with $0\le r<5$, \[ \operatorname{ex}(n,C_5,B_k)=q^{\,5-r}(q+1)^r, \] with the balanced complete $C_5$-blow-ups as the only extremal graphs. The linear order of this range is best possible since for all sufficiently large $n$, the formula fails whenever $k\ge0.14n$.

5. Close Broader context Cited by this paper Used in depth by this paper Not relevant

   Quasiplanar graphs, string graphs, and the Erdős-Gallai problem

   Jacob Fox, J. Pach, Andrew Suk · 2021 · International Symposium Graph Drawing and Network Visualization · cited 12 times · Not summarized yet

   Quasiplanar graphs, string graphs, and the Erdős-Gallai problem studies edge bounds for quasiplanar graphs alongside clique and coloring properties of string graphs.

   Abstract

   An $r$-quasiplanar graph is a graph drawn in the plane with no $r$ pairwise crossing edges. Let $s \geq 3$ be an integer and $r=2^s$. We prove that there is a constant $C$ such that every $r$-quasiplanar graph with $n \geq r$ vertices has at most $n\left(Cs^{-1}\log n\right)^{2s-4}$ edges. A graph whose vertices are continuous curves in the plane, two being connected by an edge if and only if they intersect, is called a string graph. We show that for every $\epsilon>0$, there exists $\delta>0$ such that every string graph with $n$ vertices, whose chromatic number is at least $n^{\epsilon}$ contains a clique of size at least $n^{\delta}$. A clique of this size or a coloring using fewer than $…

6. Close Broader context Not relevant

   Minimum Label $k$-Cut on Planar Graphs: Complexity and Color Structure

   Yuan-Hao Wang, Wei Wang · 2026 · Not summarized yet

   Minimum Label k-Cut studies color deletion and component counts in planar graphs, rather than crossing restrictions or edge bounds.

   Abstract

   Minimum Label $k$-Cut asks for a minimum-size set of colors such that deleting all edges of these colors leaves at least $k$ connected components. We study how the structure of the underlying graph and its color classes affects the complexity of this problem. Regarding fixed-parameter tractability with respect to $k$ on simple planar graphs, we prove that the problem is W[1]-hard, even when the underlying graph is a cactus graph and each color appears at most twice. Furthermore, for cactus graphs in which every color class is connected, we establish a complexity boundary in terms of the maximum monochromatic degree: when this degree is at most two, an optimal solution can be found in $O(Nk^2…

7. Close Broader context Not relevant

   Smallest Cubic Non-1-Planar Graphs

   S. Pupyrev · 2026 · arXiv.org · preprint, not peer reviewed · Not summarized yet

   Smallest Cubic Non-1-Planar Graphs studies crossing constraints in graph drawings, using SAT methods to determine 1-planarity of small cubic graphs.

   Abstract

   A graph is 1-planar if it has a drawing in which every edge is crossed at most once. We show that the smallest cubic non-1-planar graphs have $30$ vertices. Two such graphs are the Tutte-Coxeter graph of girth eight and a graph of girth seven that we call the Byte graph. Every subcubic graph with fewer than $30$ vertices is 1-planar. Our proof is computer-assisted, but directly testing all relevant graphs is impractical. To establish non-1-planarity of the two graphs, we extend a SAT-based solver with a custom clause propagator based on separating cycles and a case split based on graph automorphisms, allowing independent cases to be solved in parallel. To show that all smaller subcubic graph…

8. Close Another approach to the problem Cited by this paper Not relevant

   Coloring Curves that Cross a Fixed Curve

   Alexandre Rok, Bartosz Walczak · 2015 · International Symposium on Computational Geometry · cited 14 times · Not summarized yet

   Coloring Curves proves an O(n log n) edge bound for k-quasi-planar graphs when pairs of edges cross at most t times.

   Abstract

   We prove that for every integer $$t\geqslant 1$$t⩾1, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most t points is $$\chi $$χ-bounded. This is essentially the strongest $$\chi $$χ-boundedness result one can get for those kind of graph classes. As a corollary, we prove that for any fixed integers $$k\geqslant 2$$k⩾2 and $$t\geqslant 1$$t⩾1, every k-quasi-planar topological graph on n vertices with any two edges crossing at most t times has $$O(n\log n)$$O(nlogn) edges.

Prior work it builds on

1. Related Review Not relevant

   Relaxing Planarity for Topological Graphs

   J. Pach, R. Radoičić, G. Tóth · 2002 · JCDCG · cited 57 times · Not summarized yet

   Relaxing Planarity for Topological Graphs is a chapter surveying relaxed planarity concepts for topological graphs.

   Abstract

   Note: Professor Pach's number: [175]; Also in: More Sets, Graphs and Numbers, E. Gy{\Ho}ri, G.O.H. Katona, L. Lovasz (eds.), Bolyai Society mathematical studies, 15, Budapest, 2006, 559-576 Reference DCG-CHAPTER-2008-029 Record created on 2008-11-18, modified on 2017-05-12

2. Related* Same question Not relevant

   On the Maximum Number of Edges in Topological Graphs with no Four Pairwise Crossing Edges

   Eyal Ackerman · 2006 · SCG '06 · cited 99 times · Not summarized yet

   On the Maximum Number of Edges in Topological Graphs with no Four Pairwise Crossing Edges addresses an edge bound under a pairwise-crossing prohibition.

3. Related Broader context Used in depth by this paper Not relevant

   A parity Erd\H{o}s-Hajnal theorem for $t$-intersecting curves

   Andrew Suk, Su Zhou · 2026 · cited 1 time · Not summarized yet

   A parity Erdős-Hajnal theorem studies large subfamilies of pairwise t-intersecting curves with uniform intersection parity.

   Abstract

   For every fixed $t\ge 1$, we prove a parity analogue of the mighty Erd\H{o}s-Hajnal property for $t$-intersecting curves in the plane. Let $\mathcal B$ be a set of blue curves and $\mathcal G$ a set of green curves in the plane such that $\mathcal B\cup\mathcal G$ is a collection of $t$-intersecting curves in general position. We show that there exist subfamilies $\mathcal B'\subseteq\mathcal B$ and $\mathcal G'\subseteq\mathcal G$ such that $|\mathcal B'|\geq \varepsilon|\mathcal B|$ and $|\mathcal G'|\geq \varepsilon|\mathcal G|$, where $\varepsilon>0$ depends only on $t$, such that either every pair in $\mathcal B'\times\mathcal G'$ intersects an even number of times or every such pair in…

4. Related Broader context Used in depth by this paper Not relevant

   Cubical accessibility and bounds on curves on surfaces

   B. Beeker, Nir Lazarovich · 2017 · Journal of Topology · cited 6 times · Not summarized yet

   Cubical accessibility and bounds on curves on surfaces derives bounds on curves on surfaces, a setting related to the paper's surface-arc method.

   Abstract

   We bound the size of d ‐dimensional cubulations of finitely presented groups. We apply this bound to obtain acylindrical accessibility for actions on CAT(0) cube complexes and bounds on curves on surfaces.

5. Related Review Not relevant

   Notes on Geometric Graph Theory

   J. Pach · 1990 · Discrete & Computational Geometry · cited 32 times · Not summarized yet

   Notes on Geometric Graph Theory is a chapter about geometric graph theory, the broader field containing quasi-planar graph edge bounds.

   Abstract

   Note: Professor Pach's number: [092] Reference DCG-CHAPTER-2008-008 Record created on 2008-11-18, modified on 2017-05-12