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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2605.29658 |
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| _version_ | 1866918529162280960 |
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| author | Yi, Xu Yu, Gaohang |
| author_facet | Yi, Xu Yu, Gaohang |
| contents | Let $G_1$ denote the incidence graph of the complete graph $K_{q+1}$. We study limited augmented Zarankiewicz numbers in this family by combining exact 0--1 ILP computations for the smallest cases with a constructive search procedure followed by exact admissibility verification in the larger cases considered here. We obtain \[ z_L(6,4)=14,\qquad z_L(10,5)=26,\qquad z_L(15,6)\ge 43,\qquad z_L(21,7)\ge 64,\qquad z_L(28,8)\ge 88. \] The first two values are exact. The three lower bounds arise from explicitly verified admissible families with $|E_2|=13$, $|E_2|=22$, and $|E_2|=32$, respectively; the families used to obtain these bounds are nondegenerate in the sense of [8]. In each case, the resulting value improves the corresponding classical Zarankiewicz number and hence strengthens the available lower bounds for BSR(m,n) within this family. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29658 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Computational Study of Limited Augmented Zarankiewicz Numbers in the Incidence-Graph Family of Complete Graphs Yi, Xu Yu, Gaohang Combinatorics Let $G_1$ denote the incidence graph of the complete graph $K_{q+1}$. We study limited augmented Zarankiewicz numbers in this family by combining exact 0--1 ILP computations for the smallest cases with a constructive search procedure followed by exact admissibility verification in the larger cases considered here. We obtain \[ z_L(6,4)=14,\qquad z_L(10,5)=26,\qquad z_L(15,6)\ge 43,\qquad z_L(21,7)\ge 64,\qquad z_L(28,8)\ge 88. \] The first two values are exact. The three lower bounds arise from explicitly verified admissible families with $|E_2|=13$, $|E_2|=22$, and $|E_2|=32$, respectively; the families used to obtain these bounds are nondegenerate in the sense of [8]. In each case, the resulting value improves the corresponding classical Zarankiewicz number and hence strengthens the available lower bounds for BSR(m,n) within this family. |
| title | A Computational Study of Limited Augmented Zarankiewicz Numbers in the Incidence-Graph Family of Complete Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.29658 |