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The Limited Augmented Zarankiewicz Number

Liqun Qi, Chunfeng Cui, Yi Xu

Abstract

The limited augmented Zarankiewicz number $z_L(m,n)$ satisfies $\operatorname{BSR}(m,n) \ge z_L(m,n) \ge z(m,n)$, where $\operatorname{BSR}(m,n)$ is the maximum SOS rank of $m \times n$ biquadratic forms and $z(m,n)$ is the classical Zarankiewicz number. We determine the exact values of $z_L(m,n)$ for all $m,n \le 5$. In particular, we prove that $z_L(5,3) = 9$, $z_L(5,4) = 12$, and $z_L(5,5) = 14$, confirming that previously known lower bounds are tight. Moreover, by a lifting construction we obtain $z_L(6,5) \ge 17$, which is the first example where $z_L(m,n) \ge z(m,n) + 3$, demonstrating that the gap can grow with the dimensions. The analysis proceeds by enumerating all non-isomorphic extremal $C_4$-free graphs for each parameter set and systematically checking the admissible 2-edge augmentations. Our results reveal that the augmentability of a $C_4$-free graph depends critically on its specific structure, not merely on its edge count. These findings provide improved lower bounds for $\operatorname{BSR}(m,n)$ and offer a foundation for future investigations of larger parameters.

The Limited Augmented Zarankiewicz Number

Abstract

The limited augmented Zarankiewicz number satisfies , where is the maximum SOS rank of biquadratic forms and is the classical Zarankiewicz number. We determine the exact values of for all . In particular, we prove that , , and , confirming that previously known lower bounds are tight. Moreover, by a lifting construction we obtain , which is the first example where , demonstrating that the gap can grow with the dimensions. The analysis proceeds by enumerating all non-isomorphic extremal -free graphs for each parameter set and systematically checking the admissible 2-edge augmentations. Our results reveal that the augmentability of a -free graph depends critically on its specific structure, not merely on its edge count. These findings provide improved lower bounds for and offer a foundation for future investigations of larger parameters.

Paper Structure

This paper contains 52 sections, 4 theorems, 64 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Contributions of This Paper
  3. Organization of the Paper
  4. Notation
  5. Limited Augmented Zarankiewicz Number
  6. The $5 \times 3$ Case
  7. Type I.
  8. Type II.
  9. Case I.1: Two nondegenerate.
  10. Case I.2: One nondegenerate and one column-degenerate.
  11. Case I.3: One nondegenerate and one row-degenerate.
  12. Case I.4: Two column-degenerate.
  13. Case I.5: Two row-degenerate.
  14. Case I.6: One column-degenerate and one row-degenerate.
  15. Case II.1: Two nondegenerate.
  16. ...and 37 more sections

Key Result

Theorem 2.2

For all $m,n\ge 2$, $\operatorname{BSR}(m,n) \ge z_L(m,n) \ge z(m,n)$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (7)

  • Definition 2.1: Limited Augmented Zarankiewicz Number QCX26a
  • Theorem 2.2: QCX26a
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • proof