[Paper Review] A note on domination in intersecting linear systems

Adrián Vázquez-Ávila|arXiv (Cornell University)|Jun 5, 2018

Advanced Graph Theory Research8 references3 citations

TL;DR

This paper establishes that for even prime powers $ q $, any intersecting linear system of rank $ q+2 $ with domination number $ q+1 $ must arise from a spanning $ (q+1) $-uniform intersecting subsystem of the projective plane $ \Pi_q $, where the transversal number equals the 2-packing number minus one. The result characterizes extremal systems via projective plane structure and confirms a structural constraint on high-rank intersecting systems with maximal domination number.

ABSTRACT

A linear system is a pair $(P,\mathcal{L})$ where $\mathcal{L}$ is a family of subsets on a ground finite set $P$ such that $|l\cap l^\prime|\leq 1$, for every $l,l^\prime \in \mathcal{L}$. The elements of $P$ and $\mathcal{L}$ are called points and lines, respectively, and the linear system is called intersecting if any pair of lines intersect in exactly one point. A subset $D$ of points of a linear system $(P,\mathcal{L})$ is a dominating set of $(P,\mathcal{L})$ if for every $u\in P\setminus D$ there exists $v\in D$ such that $u,v\in l$, for some $l\in\mathcal{L}$. The cardinality of a minimum dominating set of a linear system $(P,\mathcal{L})$ is called domination number of $(P,\mathcal{L})$, denoted by $γ(P,\mathcal{L})$. On the other hand, a subset $R$ of lines of a linear system $(P,\mathcal{L})$ is a 2-packing if any three elements of $R$ have not a common point (are triplewise disjoint). The cardinality of a maximum 2-packing of a linear system $(P,\mathcal{L})$ is called 2-packing number of $(P,\mathcal{L})$, denoted by $ν_2(P,\mathcal{L})$. It is know for intersecting linear systems $(P,\mathcal{L})$ of rank $r$ it satisfies $γ(P,\mathcal{L})\leq r-1$. In this note we prove, if $q$ is an even prime power and $(P,\mathcal{L})$ is an intersecting linear system of rank $q+2$ satisfying $γ(P,\mathcal{L})=q+1$, then this linear system can be constructed from a spanning $(q+1)$-uniform intersecting linear subsystem $(P^\prime,\mathcal{L}^\prime)$ of the projective plane of order $q$ satisfying $τ(P^\prime,\mathcal{L}^\prime)=ν_2(P^\prime,\mathcal{L}^\prime)-1=q+1$.

Motivation & Objective

• To characterize intersecting linear systems of rank $ q+2 $ with domination number $ q+1 $, where $ q $ is an even prime power.
• To determine the structural conditions under which such systems can exist, particularly focusing on their underlying subsystems.
• To prove that such systems must be derived from a spanning $ (q+1) $-uniform intersecting subsystem of the projective plane $ \Pi_q $.
• To establish a precise relationship between the transversal number and 2-packing number in the subsystem, specifically $ \tau = \nu_2 - 1 $.

Proposed method

• Analyzes the structure of intersecting linear systems of rank $ q+2 $ with maximum domination number $ q+1 $, leveraging known bounds on domination in intersecting systems.
• Applies combinatorial bounds on 2-packings and transversals in uniform linear systems, particularly using Lemma 3.1 from prior work on even-rank systems.
• Uses the fact that a $ (q+1) $-uniform intersecting linear system with $ \nu_2 = q+2 $ and $ \tau = q+1 $ must be isomorphic to the projective plane $ \Pi_q $ when $ q $ is even.
• Employs duality and extremal combinatorics in projective planes to show that such systems are uniquely determined by their subsystems.
• Applies Lemma 2.4 and Theorem 2.1 to bound the number of lines and degrees in the subsystem, ensuring consistency with $ \Pi_q $.
• Uses the pigeonhole principle and linearity constraints to rule out alternative constructions beyond those derived from $ \Pi_q $.

Experimental results

Research questions

• RQ1 Under what conditions can an intersecting linear system of rank $ q+2 $ achieve a domination number of $ q+1 $, where $ q $ is an even prime power?
• RQ2 What structural properties must the $ (q+1) $-uniform spanning subsystem of such a system possess?
• RQ3 Can such systems be fully characterized by their relationship to the projective plane $ \Pi_q $?
• RQ4 Is it possible for a system with $ \gamma = q+1 $ to exist outside of constructions derived from $ \Pi_q $, and if so, under what constraints?
• RQ5 What is the precise relationship between the transversal number and 2-packing number in the subsystems of such extremal systems?

Key findings

• For every intersecting linear system $ (P,\mathcal{L}) $ of rank $ q+2 $ with $ \gamma(P,\mathcal{L}) = q+1 $, where $ q $ is an even prime power, the system arises from a spanning $ (q+1) $-uniform intersecting subsystem $ (P^\prime,\mathcal{L}^\prime) $ of the projective plane $ \Pi_q $.
• The subsystem $ (P^\prime,\mathcal{L}^\prime) $ satisfies $ \tau(P^\prime,\mathcal{L}^\prime) = \nu_2(P^\prime,\mathcal{L}^\prime) - 1 = q+1 $, confirming a tight duality between transversal and 2-packing parameters.
• The number of lines in $ \mathcal{L}^\prime $ is at least $ 3q $, and when $ |\mathcal{L}^\prime| = 3q $, the 2-packing number is exactly $ q+2 $, matching the maximum possible for such systems.
• When $ q $ is even, the 2-packing number $ \nu_2(P^\prime,\mathcal{L}^\prime) = q+2 $, which equals the size of an oval in $ \Pi_q $, confirming the extremality of the subsystem.
• The system $ (P^\prime,\mathcal{L}^\prime) $ is isomorphic to $ \Pi_q $, the projective plane of order $ q $, due to its line count, uniformity, and extremal parameters.
• The result generalizes earlier findings for rank 4 systems, where such systems were shown to be constructed from the Fano plane, now extended to all even prime power orders.