Incidence Rank and Bounded Defect for Linear Hypergraphs with Cograph Line Graphs
Abstract
Let N(H) be the edge--vertex incidence matrix of a finite linear s-uniform hypergraph whose line graph is a cograph, and define Rs(H)=(s+1)rankℝN(H)−s|E(H)|. The inequality Rs(H)≥0 holds, with equality characterized by affine planes of order s on every row-dependent component. Under the corresponding finite-net completion hypothesis, connected systems with bounded Rs admit an explicit classification. If F is connected, Rs(F)=d≤ s−1, the completion hypothesis holds through deficiency d, and s>(d−1)2, then either N(F) has full row rank and |E(F)|=d, or F lies in a unique affine plane of order s. In the latter case, for some 0≤ h≤ d, the system contains every line in s+1−h parallel classes and exactly d−h additional lines from the remaining h classes. Conversely, each such system has defect d. Bruck's theorem gives the classification for every fixed d and all sufficiently large s, while Metsch's completion theorem covers d=o(s1/3). As a global consequence, if the total defect is at most D in the completion range, deleting at most D edges leaves a vertex-disjoint union of finite nets and affine planes. The associated column defect and row leverage are determined explicitly as well.
Related
- Cyclic Incidence Orderings of Complete Graphs and 3-Uniform Hypergraphs
- Ordered Ramsey numbers of 3-uniform hypergraphs with bounded weak degeneracy
- Turán problems with bounded matching number in k-uniform hypergraphs
- General-position sets and linear hypergraphs
- The upper bounds on occupancy fraction and variance fraction for linear regular uniform hypergraphs
BibTeX
@misc{ramani2026incidence,
title = {{Incidence Rank and Bounded Defect for Linear Hypergraphs with Cograph Line Graphs}},
author = {Mahesh Ramani},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27777},
eprint = {2609.27777},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27777},
}
RIS
TY - UNPB
TI - Incidence Rank and Bounded Defect for Linear Hypergraphs with Cograph Line Graphs
AU - Mahesh Ramani
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27777
UR - https://arxiv.org/abs/2609.27777
AB - Let N(H) be the edge--vertex incidence matrix of a finite linear s-uniform hypergraph whose line graph is a cograph, and define R_s(H)=(s+1)rank_ℝN(H)−s|E(H)|. The inequality R_s(H)≥0 holds, with equality characterized by affine planes of order s on every row-dependent component. Under the corresponding finite-net completion hypothesis, connected systems with bounded R_s admit an explicit classification. If F is connected, R_s(F)=d≤ s−1, the completion hypothesis holds through deficiency d, and s>(d−1)^2, then either N(F) has full row rank and |E(F)|=d, or F lies in a unique affine plane of order s. In the latter case, for some 0≤ h≤ d, the system contains every line in s+1−h parallel classes and exactly d−h additional lines from the remaining h classes. Conversely, each such system has defect d. Bruck's theorem gives the classification for every fixed d and all sufficiently large s, while Metsch's completion theorem covers d=o(s^1/3). As a global consequence, if the total defect is at most D in the completion range, deleting at most D edges leaves a vertex-disjoint union of finite nets and affine planes. The associated column defect and row leverage are determined explicitly as well.
ER -