Applications of a new separator theorem for string graphs

An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphs G with n vertices: (i) if K_t is not a subgraph of G for some t, then the chromatic number of G is at most (\log n)^{O(\log t)}; (ii) if K_{t,t} is not a subgraph of G, then G has at most t(\log t)^{O(1)}n edges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdos-Hajnal conjecture almost holds for string graphs.Comment: 7 page

On the size of planarly connected crossing graphs

We prove that if an $n$-vertex graph $G$ can be drawn in the plane such that each pair of crossing edges is independent and there is a crossing-free edge that connects their endpoints, then $G$ has $O(n)$ edges. Graphs that admit such drawings are related to quasi-planar graphs and to maximal $1$-planar and fan-planar graphs.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Applications of a New Separator Theorem for String Graphs

An intersection graph of curves in the plane is called a string graph. Matoušek almost completely settled a conjecture of the authors by showing that every string graph with m edges admits a vertex separator of size $O(\sqrt{m}\log m)$ . In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphs G with n vertices: (i) if Kt ⊈ G for some t, then the chromatic number of G is at most (log n) O(log t); (ii) if Kt,t ⊈ G, then G has at most t(log t) O(1) n edges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdős-Hajnal conjecture almost holds for string graph

On the Number of Edges of Fan-Crossing Free Graphs

A graph drawn in the plane with n vertices is k-fan-crossing free for k > 1 if there are no k+1 edges $g,e_1,...e_k$, such that $e_1,e_2,...e_k$ have a common endpoint and $g$ crosses all $e_i$. We prove a tight bound of 4n-8 on the maximum number of edges of a 2-fan-crossing free graph, and a tight 4n-9 bound for a straight-edge drawing. For k > 2, we prove an upper bound of 3(k-1)(n-2) edges. We also discuss generalizations to monotone graph properties

On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs

Fan-planar graphs were recently introduced as a generalization of 1-planar graphs. A graph is fan-planar if it can be embedded in the plane, such that each edge that is crossed more than once, is crossed by a bundle of two or more edges incident to a common vertex. A graph is outer-fan-planar if it has a fan-planar embedding in which every vertex is on the outer face. If, in addition, the insertion of an edge destroys its outer-fan-planarity, then it is maximal outer-fan-planar. In this paper, we present a polynomial-time algorithm to test whether a given graph is maximal outer-fan-planar. The algorithm can also be employed to produce an outer-fan-planar embedding, if one exists. On the negative side, we show that testing fan-planarity of a graph is NP-hard, for the case where the rotation system (i.e., the cyclic order of the edges around each vertex) is given

The number of edges in k-quasi-planar graphs

A graph drawn in the plane is called k-quasi-planar if it does not contain k pairwise crossing edges. It has been conjectured for a long time that for every fixed k, the maximum number of edges of a k-quasi-planar graph with n vertices is O(n). The best known upper bound is n(\log n)^{O(\log k)}. In the present note, we improve this bound to (n\log n)2^{\alpha^{c_k}(n)} in the special case where the graph is drawn in such a way that every pair of edges meet at most once. Here \alpha(n) denotes the (extremely slowly growing) inverse of the Ackermann function. We also make further progress on the conjecture for k-quasi-planar graphs in which every edge is drawn as an x-monotone curve. Extending some ideas of Valtr, we prove that the maximum number of edges of such graphs is at most 2^{ck^6}n\log n.Comment: arXiv admin note: substantial text overlap with arXiv:1106.095

Coloring curves that cross a fixed curve

We prove that for every integer $t\geq 1$, 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 $\chi$-bounded. This is essentially the strongest $\chi$-boundedness result one can get for this kind of graph classes. As a corollary, we prove that for any fixed integers $k\geq 2$ and $t\geq 1$, every $k$-quasi-planar topological graph on $n$ vertices with any two edges crossing at most $t$ times has $O(n\log n)$ edges.Comment: Small corrections, improved presentatio