Unavoidable patterns and plane paths in dense topological graphs

TL;DR

The paper investigates unavoidable patterns in dense simple topological graphs, focusing on plane paths and bipartite configurations. It proves that large complete bipartite simple topological graphs must contain a subgraph weakly isomorphic to $C_{k,k}$ when $|U|>2(k-1)^4$ and $|V|ge 2^{k^{5k}}$, and derives corresponding edge-density bounds for graphs with no plane path of length $k$, including a sharper bound for $k=3$. It introduces and analyzes local thrackles, providing a complete geometric classification for straight-line drawings, bounding edge counts for $x$-monotone local thrackles, and exploring general local thrackles via forbidden substructures. Finally, it proves a near-linear bound for graphs with no self-intersecting path of length 4 and discusses related Ramsey-type questions and open problems, tying together planarity arguments, decomposition lemmas, and subdivisions in a cohesive extremal framework.

Abstract

Let $C_{s,t}$ be the complete bipartite geometric graph, with $s$ and $t$ vertices on two distinct parallel lines respectively, and all $s t$ straight-line edges drawn between them. In this paper, we show that every complete bipartite simple topological graph, with parts of size $2(k-1)^4 + 1$ and $2^{k^{5k}}$, contains a topological subgraph weakly isomorphic to $C_{k,k}$. As a corollary, every $n$-vertex simple topological graph not containing a plane path of length $k$ has at most $O_k(n^{2 - 8/k^4})$ edges. When $k = 3$, we obtain a stronger bound by showing that every $n$-vertex simple topological graph not containing a plane path of length 3 has at most $O(n^{4/3})$ edges. We also prove that $x$-monotone simple topological graphs not containing a plane path of length 3 have at most a linear number of edges.

Figures (10)

• Figure 1: A drawing of each of the $5$ types of $C_4$'s.
• Figure 2: Proof of transitivity of $G^{(1)}_{s,t}$.
• Figure 3: Two forbidden subgraphs $G_1$ (left) and $G_2$ (right) for geometric local thrackles.
• Figure 4: Even cycles with spikes can be drawn as a geometric local thrackle.
• Figure 5: Two paths (black and red) joined by additional $2$-paths (blue) drawn as a local thrackle with $x$-monotone edges.

Theorems & Definitions (43)

• Theorem 1.1
• Proposition 1.2
• Theorem 1.3
• Corollary 1.4
• Proposition 1.5
• Proposition 1.6
• Proposition 1.7
• Theorem 1.8
• Lemma 2.1
• Lemma 2.2