The Limited Augmented Zarankiewicz Number

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 paper introduces the limited augmented Zarankiewicz number to refine lower bounds on SOS ranks by incorporating controlled 2-edge augmentations in bipartite graphs.
• It employs exhaustive combinatorial enumeration and algorithmic verification for small dimensions (m,n ≤ 5) to demonstrate explicit tight bounds and structural constraints.
• Results extend to larger configurations (e.g., 6×5), revealing a growing gap between classical and augmented bounds, with implications for complexity in sum-of-squares decompositions.

The Limited Augmented Zarankiewicz Number and Its Role in Biquadratic SOS Ranks

Background and Motivation

The study centers on the maximal number of squares required to represent an SOS (sum of squares) biquadratic form on $\mathbb{R}^m \otimes \mathbb{R}^n$. Specifically, for an $m \times n$ real biquadratic form

$$P(\boldsymbol{x}, \boldsymbol{y}) = \sum_{i,k=1}^{m}\sum_{j,l=1}^{n} a_{ijkl} x_i x_k y_j y_l,$$

one asks for the minimal $r$ such that $P$ can be written as the sum of $r$ squares of bilinear forms. The supremum of this value over all SOS $m \times n$ biquadratic forms is denoted $\operatorname{BSR}(m, n)$.

A deep correspondence was established between lower bounds for $\operatorname{BSR}(m, n)$ and extremal combinatorics, specifically the Zarankiewicz number $z(m, n)$. The latter is the maximum number of edges in an $m \times n$0 bipartite graph containing no $m \times n$1 (i.e., no $m \times n$2-cycle). This connection, established in [CQX26], provides the universal bound $m \times n$3. However, explicit calculation uncovered strict inequality in several small-dimension instances.

This paper introduces a refined combinatorial parameter, the limited augmented Zarankiewicz number $m \times n$4, which exactly tracks the capabilities of augmenting extremal $m \times n$5-free bipartite graphs by a restricted class of "2-edges" corresponding to cross-square terms, mapping exactly to additional sum-of-squares terms in biquadratic form decompositions. The key structural constraints are formalized by a set of forbidden substructures, generalizing $m \times n$6, to ensure correspondence with irreducible SOS expressions.

Core Definitions and Theoretical Framework

Limited Augmented Bipartite Graphs

A limited augmented bipartite graph $m \times n$7 is built on vertex classes of sizes $m \times n$8 and $m \times n$9 and consists of standard edges (1-edges, $$P(\boldsymbol{x}, \boldsymbol{y}) = \sum_{i,k=1}^{m}\sum_{j,l=1}^{n} a_{ijkl} x_i x_k y_j y_l,$$0) as well as 2-edges ($$P(\boldsymbol{x}, \boldsymbol{y}) = \sum_{i,k=1}^{m}\sum_{j,l=1}^{n} a_{ijkl} x_i x_k y_j y_l,$$1), which are unordered pairs of cells with constraints: 1-edges must form a $$P(\boldsymbol{x}, \boldsymbol{y}) = \sum_{i,k=1}^{m}\sum_{j,l=1}^{n} a_{ijkl} x_i x_k y_j y_l,$$2-free bipartite graph; 2-edges cannot share either cell (simplicity); and their inclusion is forbidden if they generate a generalized $$P(\boldsymbol{x}, \boldsymbol{y}) = \sum_{i,k=1}^{m}\sum_{j,l=1}^{n} a_{ijkl} x_i x_k y_j y_l,$$3-cycle as defined by three explicit local conditions (including occupation of opposite cells and specialized five-cell patterns).

The limited augmented Zarankiewicz number $$P(\boldsymbol{x}, \boldsymbol{y}) = \sum_{i,k=1}^{m}\sum_{j,l=1}^{n} a_{ijkl} x_i x_k y_j y_l,$$4 is the maximum total size $$P(\boldsymbol{x}, \boldsymbol{y}) = \sum_{i,k=1}^{m}\sum_{j,l=1}^{n} a_{ijkl} x_i x_k y_j y_l,$$5 of such a graph.

Biquadratic SOS Rank Link

For any $$P(\boldsymbol{x}, \boldsymbol{y}) = \sum_{i,k=1}^{m}\sum_{j,l=1}^{n} a_{ijkl} x_i x_k y_j y_l,$$6, it follows that

$$P(\boldsymbol{x}, \boldsymbol{y}) = \sum_{i,k=1}^{m}\sum_{j,l=1}^{n} a_{ijkl} x_i x_k y_j y_l,$$7

with equality known in several cases. Hence, $$P(\boldsymbol{x}, \boldsymbol{y}) = \sum_{i,k=1}^{m}\sum_{j,l=1}^{n} a_{ijkl} x_i x_k y_j y_l,$$8 is a refined lower bound on the true biquadratic SOS rank, and tightness is a central research theme.

Enumeration and Combinatorial Analysis for Small Dimensions ($$P(\boldsymbol{x}, \boldsymbol{y}) = \sum_{i,k=1}^{m}\sum_{j,l=1}^{n} a_{ijkl} x_i x_k y_j y_l,$$9)

The major technical innovation of the work is an exhaustive combinatorial enumeration for all extremal graphs on $r$0 vertices with no $r$1, together with a systematic algorithmic verification of possible 2-edge augmentations in accordance with the imposed structural conditions. This is carried out with full explicitness for all pairs $r$2.

Strong numerical results:

• $r$3, with only one 2-edge augment possible for either of the two isomorphism classes of extremal $r$4-free graphs.
• $r$5, with three extremal $r$6-free configurations analyzed; only one (Type C) admits two 2-edges (one nondegenerate and one column-degenerate), the others only one.
• $r$7, with two non-isomorphic extremal $r$8-free graphs; only one allows two nondegenerate 2-edges, the other only one.

A detailed summary table:

$r$9	$P$0	$P$1
(3,3)	6	6
(4,3)	7	8
(4,4)	9	10
(5,3)	8	9
(5,4)	10	12
(5,5)	12	14

For each, the methods involve complete casework on all candidate 2-edge additions, ensuring none of the structural forbidden patterns are created.

Contradictory/Notable claims:

• The gap $P$2 is strictly positive for many cases, showing that classical extremal combinatorial tools fail to capture the algebraic complexity of biquadratic SOS rank.
• The gap for small parameter pairs is bounded ($P$3 at most up to $P$4). This structure-sensitive behavior underlines the combinatorial subtlety of augmentability: only special extremal graphs are highly augmentable.

Extension to Larger Parameters ($P$5) and Growth of the Gap

The authors employ a lifting strategy, extending the optimal $P$6 configuration to $P$7 by adding rows, columns, and carefully selected edges. They construct a $P$8-free $P$9 bipartite graph with $r$0 1-edges, augmented by three 2-edges while maintaining all forbidden substructure constraints, thereby establishing that

$r$1

This is the first explicit demonstration that the $r$2 gap can attain three, and by extension potentially larger values as dimensions grow. This contradicts any conjecture that the gap is universally $r$3 or asymptotically bounded in $r$4.

Graph Structure Sensitivity and Augmentability

A key theoretical outcome is that augmentability is a property not solely of $r$5 but of specific global configurations (isomorphism type). This is transparent in the $r$6 and $r$7 cases, where only certain extremal graph structures admit the maximum possible number of 2-edges without producing forbidden cycles.

This observation has implications for algebraic optimization: the existence of tight SOS rank constructions is not a function only of the count of "nearly complete" bipartite graphs, but their precise structural arrangement.

Implications, Open Problems, and Future Directions

On both practical and theoretical planes, these results recalibrate understanding of the sum-of-squares decomposition problem for biquadratic forms:

• Practical implication: Improved lower bounds for SOS ranks, which translate into worst-case complexity metrics for symbolic and numerical semidefinite programming relaxations in polynomial optimization, particularly for forms with extremal monomial support patterns.
• Theoretical implication: The limited augmented Zarankiewicz number is poised to play an intermediary role between algebraic geometry (SOS ranks, Waring decompositions) and extremal combinatorics (forbidden bipartite subgraphs), suggesting further development of hybridized theory.

Notable open questions presented:

• Exact value of $r$8 (tightness of the constructed lower bound remains unresolved).
• Growth rate of the gap $r$9: is it always sublinear, polylogarithmic, or could it be unbounded as $m \times n$0?
• For which $m \times n$1 does $m \times n$2, and is this universally true?

Potential AI relevance: Understanding the combinatorial underpinnings of tight SOS decompositions is central to the worst-case analysis of polynomial optimization and quantifier elimination algorithms (often invoked implicitly in algebraic geometry tools for constraint satisfaction). Practical advances in SDP-solving for polynomial systems may benefit directly from sharp lower bounds on required sum-of-squares terms.

Conclusion

This work rigorously determines the limited augmented Zarankiewicz number for all pairs $m \times n$3, demonstrating that previously posited lower bounds are exact in every such case, and inaugurating a new family of sharp lower bounds for larger values (notably $m \times n$4), where the gap with the classical Zarankiewicz number becomes strictly larger. The combinatorics of augmentability are shown to be structure-sensitive, with implications for both algebraic and combinatorial extremal theory, as well as symbolic algebra and optimization. These findings delineate several pathways for further study, particularly regarding asymptotic growth rates, structural characterization of maximally augmentable graphs, and potential equality with the true biquadratic SOS rank (2604.04111).