发表机构
School of Mathematical Sciences, Chongqing Normal University（重庆师范大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文基于 $K_{5t}$ 关联二部图构造新的无限图族，推导出受限增广 Zarankiewicz 数的二次下界，其相对间隙渐近收敛于 $2/5$，优于已知的 $1/4$，并验证了 $t=1$ 时的精确值，扩展了极值二部图库并加强了双二次 SOS 秩下界。
AI 中文摘要
受限增广 Zarankiewicz 数 \\(z_L(m,n)\\) 为双二次型的最大 SOS 秩提供了核心组合下界。所有先前已知的无限可容许图族依赖于 \\(K_{4t}\\) 关联图，达到渐近相对间隙极限 \\(1/4\\)。本文开发了一个新的无限族，由具有 \\(\mathbb{Z}_5\\) 循环标记的块划分的 \\(K_{5t}\\) 关联二部图构成。我们构造了有效的非退化块内和块间 2-边，推导出 \\(z_L\\) 的二次闭式下界，并证明其相对间隙渐近收敛于 \\(2/5\\)。对 \\(t=1\\) 的完整枚举验证了精确值 \\(z_L(10,5)=23\\)。在非退化和广义 \\(C_4\\)-自由约束下，\\(2/5\\) 被证明是该块框架下可达到的最大渐近比率。我们的结果扩展了极值二部图的库，并加强了双二次 SOS 秩的下界，最后概述了针对一般 \\(K_{kt}\\) 构造的进一步开放问题。
英文摘要
The restricted augmented Zarankiewicz number \(z_L(m,n)\) yields core combinatorial lower bounds for the maximal SOS rank of biquadratic forms. All previously known infinite admissible graph families rely on \(K_{4t}\) incidence graphs, attaining an asymptotic relative gap limit of \(1/4\). This work develops a new infinite family built from \(K_{5t}\) incidence bipartite graphs with \(\mathbb{Z}_5\) cyclic labeling for block partitions. We construct valid nondegenerate intra-block and inter-block 2-edges, derive a quadratic closed-form lower bound of \(z_L\), and prove its relative gap converges asymptotically to \(2/5\). Full enumeration for \(t=1\) verifies the exact value \(z_L(10,5)=23\). Under nondegenerate and generalized \(C_4\)-free constraints, the ratio \(2/5\) is shown to be the maximal asymptotic ratio attainable under this block framework. Our results expand the library of extremal bipartite graphs and sharpen lower bounds for biquadratic SOS rank, with further open problems for general \(K_{kt}\) constructions outlined in closing.
Comments19 pages
