Unavoidable patterns and plane paths in dense topological graphs

2025-12-04

AI Summary

This study addresses two fundamental extremal problems in topological graph theory. First, it investigates unavoidable structural patterns in dense topological graphs and their sparsity thresholds: it proves that every sufficiently large complete bipartite topological graph contains a weakly isomorphic copy of the complete bipartite geometric graph $C_{k,k}$. Second, it determines the maximum number of edges in a simple topological graph on $n$ vertices containing no $k$-edge plane path. Methodologically, the work integrates tools from geometric graph theory, extremal graph theory, weak isomorphism analysis, and $x$-monotonicity constraints. Its key contribution is the first tight extremal bound for topological graphs avoiding long plane paths—specifically, an $O(n^{4/3})$ upper bound for $k=3$, matched by a linear lower bound. These results establish a precise quantitative link between planar path avoidance and graph sparsity, significantly advancing the extremal theory of topological graphs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Determines edge bounds for topological graphs without plane paths
• Establishes unavoidable patterns in dense bipartite topological graphs
• Proves linear edge bounds for x-monotone graphs avoiding length-3 paths

Innovation

• Complete bipartite topological graph containment condition
• Edge bound for graphs without plane paths
• Linear edges for x-monotone graphs without length-3 paths