Quasiplanar graphs, string graphs, and the Erdős–Gallai problem
An r-quasiplanar graph is a graph drawn in the plane with no r pairwise crossing edges. Let s≥3 be an integer and r=2s. We prove that there is a constant C such that every r-quasiplanar graph with n≥r vertices has at most nCs−1logn2s−4 edges. A graph whose vertices are continuous curves in the plane...
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| Veröffentlicht in: | European journal of combinatorics Jg. 119; S. 103811 |
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| ISSN: | 0195-6698 |
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| Abstract | An r-quasiplanar graph is a graph drawn in the plane with no r pairwise crossing edges. Let s≥3 be an integer and r=2s. We prove that there is a constant C such that every r-quasiplanar graph with n≥r vertices has at most nCs−1logn2s−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 ϵ>0, there exists δ>0 such that every string graph with n vertices whose chromatic number is at least nϵ contains a clique of size at least nδ. A clique of this size or a coloring using fewer than nϵ colors can be found by a polynomial time algorithm in terms of the size of the geometric representation of the set of strings.
In the process, we use, generalize, and strengthen previous results of Lee, Tomon, and others. All of our theorems are related to geometric variants of the following classical graph-theoretic problem of Erdős, Gallai, and Rogers. Given a Kr-free graph on n vertices and an integer s<r, at least how many vertices can we find such that the subgraph induced by them is Ks-free? |
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| AbstractList | An r-quasiplanar graph is a graph drawn in the plane with no r pairwise crossing edges. Let s≥3 be an integer and r=2s. We prove that there is a constant C such that every r-quasiplanar graph with n≥r vertices has at most nCs−1logn2s−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 ϵ>0, there exists δ>0 such that every string graph with n vertices whose chromatic number is at least nϵ contains a clique of size at least nδ. A clique of this size or a coloring using fewer than nϵ colors can be found by a polynomial time algorithm in terms of the size of the geometric representation of the set of strings.
In the process, we use, generalize, and strengthen previous results of Lee, Tomon, and others. All of our theorems are related to geometric variants of the following classical graph-theoretic problem of Erdős, Gallai, and Rogers. Given a Kr-free graph on n vertices and an integer s<r, at least how many vertices can we find such that the subgraph induced by them is Ks-free? |
| ArticleNumber | 103811 |
| Author | Pach, János Suk, Andrew Fox, Jacob |
| Author_xml | – sequence: 1 givenname: Jacob surname: Fox fullname: Fox, Jacob email: jacobfox@stanford.edu organization: Stanford University, Stanford, CA, United States of America – sequence: 2 givenname: János surname: Pach fullname: Pach, János email: pach@cims.nyu.edu organization: Rényi Institute of Mathematics, POB 127, H-1364 Budapest, Hungary – sequence: 3 givenname: Andrew surname: Suk fullname: Suk, Andrew email: asuk@ucsd.edu organization: Department of Mathematics, University of California at San Diego, La Jolla, CA, 92093, United States of America |
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| Snippet | An r-quasiplanar graph is a graph drawn in the plane with no r pairwise crossing edges. Let s≥3 be an integer and r=2s. We prove that there is a constant C... |
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| Title | Quasiplanar graphs, string graphs, and the Erdős–Gallai problem |
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