On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers
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| Format: | Preprint |
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2025
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| author | Chalermsook, Parinya Orgo, Ly Zarsav, Minoo |
| author_facet | Chalermsook, Parinya Orgo, Ly Zarsav, Minoo |
| contents | This paper considers the \textit{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, $Z(n;k) \in Ω\left(n k \cdot \frac{\log n}{\log \log n}\right)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_20737 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers Chalermsook, Parinya Orgo, Ly Zarsav, Minoo Combinatorics Data Structures and Algorithms This paper considers the \textit{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, $Z(n;k) \in Ω\left(n k \cdot \frac{\log n}{\log \log n}\right)$ 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. |
| title | On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers |
| topic | Combinatorics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2510.20737 |