A Weak Condition for Limited Augmented Zarankiewicz Numbers

This paper introduces the weak augmented Zarankiewicz number zwA(m,n) and the weak limited augmented Zarankiewicz number zwL(m,n), which relax the admissibility conditions for augmented bipartite graphs. We show that the weak framework still guarantees irreducibility of the associated doubly simple biquadratic forms, with SOS rank equal to the total number of edges. This yields BSR(m,n)≥zwA(m,n)≥zwL(m,n)≥zL(m,n)≥z(m,n). We give three complementary strict improvements. A 5×3 weak admissible limited augmented graph with two row-degenerate 2-edges gives zwL(5,3)≥10>9=zL(5,3), and hence BSR(5,3)≥10. A 6×3 weak admissible limited augmented graph with a mutually complementary pair of nondegenerate 2-edges gives zwL(6,3)≥12>11=zL(6,3), and hence BSR(6,3)≥12. A 15×6 construction on the incidence graph of K6 with 14 nondegenerate 2-edges gives zwL(15,6)≥44>43. Finally, a 4×4 reducible example shows that (W3), like (W2′), cannot be omitted.