Quasiplanar Graphs, String Graphs, and the Erdos-Gallai Problem

TL;DR

This paper establishes bounds on edges in r-quasiplanar graphs, explores properties of string graphs related to chromatic number and clique size, and connects these results to classical extremal graph theory problems.

Contribution

It provides new bounds for r-quasiplanar graphs and advances understanding of string graphs' chromatic and clique properties, generalizing previous results.

Findings

r-quasiplanar graphs have at most n(Cs^{-1} log n)^{2s-4} edges

string graphs with high chromatic number contain large cliques

polynomial time algorithms can find large cliques or colorings in 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 $n^{\epsilon}$ 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 Erdos, Gallai, and Rogers. Given a $K_r$-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 $K_s$-free?