On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers

2025-10-23

AI Summary

This paper investigates the Zarankiewicz problem for bipartite graphs of low Ferrers dimension—i.e., maximizing the number of edges while forbidding a $K_{k,k}$ subgraph. Focusing on Ferrers dimensions 3 and 4, we establish the first phase-transition phenomenon: edge bounds are linear in $n$ for dimension 3, whereas dimension 4 triggers a sharp complexity jump. Our approach integrates extremal graph theory, combinatorial geometry, and Ferrers-structure decomposition, augmented by a novel bichromatic diagonal argument. We derive tight asymptotic upper bounds: $2n(k-1)$ for chordal bipartite graphs and $54n(k-1)$ for grid intersection graphs—substantially improving prior exponential dependencies of the form $O(2^{O(k)}n)$. This work provides the first fine-grained threshold analysis of extremal behavior driven by Ferrers dimension and establishes optimal asymptotic orders for the Zarankiewicz problem on chordal bipartite and intersection graph classes.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

This paper considers the extit{Zarankiewicz problem} in graphs with low-dimensional geometric representation (i.e., low Ferrers dimension). Our first result reveals a separation between bipartite graphs of Ferrers dimension three and four: while $Z(n;k) leq 9n(k-1)$ for graphs of Ferrers dimension three, $Z(n;k) in Omegaleft(n k cdot frac{log n}{log log n} ight)$ for Ferrers dimension four graphs (Chan&Har-Peled, 2023) (Chazelle, 1990). To complement this, we derive a tight upper bound of $2n(k-1)$ for chordal bigraphs and $54n(k-1)$ for grid intersection graphs (GIG), a prominent graph class residing in four Ferrers dimensions and capturing planar bipartite graphs as well as bipartite intersection graphs of rectangles. Previously, the best-known bound for GIG was $Z(n;k) in O(2^{O(k)} n)$, implied by the results of Fox&Pach (2006) and Mustafa&Pach (2016). Our results advance and offer new insights into the interplay between Ferrers dimensions and extremal combinatorics.

Problem

• Studies Zarankiewicz problem in low-dimensional geometric graphs
• Establishes separation between Ferrers dimension three and four
• Provides tight bounds for chordal bigraphs and grid intersection graphs

Innovation

• Establishes separation between Ferrers dimension three and four
• Derives tight upper bounds for chordal bigraphs
• Improves bounds for grid intersection graphs significantly