On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers

TL;DR

The paper tackles the Zarankiewicz problem for geometric bipartite graphs, focusing on how Ferrers dimension affects extremal edge counts. It introduces a dichotomy between Ferrers dimension three and four: CHAIN^3 graphs admit tight linear-in-n upper bounds $Z(n;k) \le 9n(k-1)$, while CHAIN^4 graphs admit a nearly linear lower bound $\Omega\left(nk \frac{\log n}{\log \log n}\right)$, complemented by a general upper bound $Z_{{CHAIN}}^d(n;k) = O(nk \lceil\log n\rceil^{d-3})$ for $d\ge3$. The study tightens bounds for prominent four-dimensional classes, proving $Z_{{GIG}}(n;k) \le 54n(k-1)$ and, more generally, $Z_{{GIG}}(m,n;k,k) \le 54(m+n)(k-1)$ via a 27(k-1)-degeneracy argument using charging schemes. It also provides a tight bound for chordal bipartite graphs, $Z_{{\mathcal{C}}}(n;k) \le 2n(k-1)$. Collectively, these results clarify the influence of Ferrers dimension on extremal growth in geometric intersection graphs and connect to prior work by Chan–Har–Peled and others, with algorithmic consequences for detecting large bicliques when exceeding bounds.

Abstract

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.

Figures (10)

• Figure 1: The relation between graph classes considered in this paper. All inclusions denoted by arrows are known to be proper. Our results imply the separation shown by the dotted curve. We show that $\mathop{\mathrm{\sf{ CHAIN}}}\nolimits^3$, $\mathop{\mathrm{\sf{ GIG}}}\nolimits$, and chordal bipartite graphs satisfy $Z_{{\mathcal{C}}}(n;k) \in O(nk)$, which implies the same for the rest of the graph classes below them.
• Figure 4: The scenarios that define relations in ${\mathcal{P}}^{DL}$ (two leftmost), ${\mathcal{P}}^{DR}$ (two in the middle) and ${\mathcal{P}}^{C}$ (two rightmost images), where $s'$ succeeds $s$.
• Figure 5: The edge $\{u, v\}$ is DL-bulky, if $\phi(v)$ (purple) contains $k-1$ segments succeeding $\phi(u)$ (green) in ${\mathcal{P}}^{DL}$.
• Figure 6: down-heavy segments are represented by the green vertical segments on the left and up-heavy segments by the purple ones on the right.
• Figure 7: \ref{['alg:close_type']} credit receivers are shown in purple and \ref{['alg:estranged_type']} credit receivers in green.

Theorems & Definitions (39)

• Theorem 1
• Corollary 2
• Theorem 3
• Theorem 4
• Proposition 5
• Theorem 6
• Theorem 8: klinz1995permutingDBLP:journals/dm/Farber83
• Lemma 9
• Theorem 10
• Theorem 10