Three-Edges and the SOS Rank of Biquadratic Forms: Extending the Augmented Zarankiewicz Framework

Abstract: The limited augmented Zarankiewicz number $z_L(m,n)$ corresponds to 2-edges $(i,j;k,l)$ in a $C_4$-free bipartite graph, each representing a square $(x_i y_j + x_k y_l)^2$. We introduce \emph{3-edges} $(i,j;k,l;p,q)$ representing $(x_i y_j + x_k y_l + x_p y_q)^2$, and define the numbers $z_{3L}(m,n)$ and $z_{3A}(m,n)$ by forbidding generalized $C_4$ cycles. We prove that for any 3-edge-augmented graph without such cycles, the corresponding doubly simple biquadratic form has SOS rank equal to the total number of edge contributions. As applications, we show $z_{3L}(5, 3) = 10$, $z_{3L}(6,4) \ge 16$ and $z_{3L}(5,5) \ge 16$, improving the known bounds $z_L(5, 3) = 9$, $z_L(6,4)=14$ and $z_L(5,5)=14$. The constructions in the $5 \times 5$ and $6 \times 4$ cases are naturally explained as 3-edges, providing a unified combinatorial framework for SOS rank lower bounds beyond the limited augmented Zarankiewicz number.