[논문 리뷰] Skirting the $n$-tuples

Sam Adriaensen, Ferdinand Ihringer외 2명|arXiv (Cornell University)|2026. 02. 01.

Advanced Graph Theory Research인용 수 0

한 줄 요약

본 논문은 해밍 거리 하에서 𝑍𝑞^𝑛의 가장 작은 skirting 집합을 연구하고 이를 특정 그래프의 총 우세 집합(total dominating sets)과 동일시하며, 존재하는 지수 증가율 f(n,q) ~ Cq^( (1+o(1))n )를 경계 및 구성과 함께 보인다.

ABSTRACT

Let $n\ge 2$ and $q\ge 2$ be given. The set $X = \mathbb Z_q^n$ is a metric space of diameter $n$ under the Hamming metric $d(\cdot,\cdot)$. We seek a smallest set $S\subseteq X$ that ``skirts'' every $q$-ary $n$-tuple in the sense that every $x\in X$ is at distance $n$ from at least one element of $S$. Thus we aim to compute the total domination number $f(n,q)$ of the graph $G(n,q)$ with vertex set $X$ and edge set $\{ xy \, \| \, d(x,y)=n\}$. We provide constructions and bounds for this number, establishing $f(n,q) = C_q^{(1+o(1))n}$ for some constants $2=C_2>C_3 \geq \cdots$ which we are only able to estimate at the present time.

연구 동기 및 목표

• Define the skirting set concept in Zq^n under the Hamming metric and relate it to total domination in G(n,q).
• Establish that f(n,q) grows exponentially with n and identify the asymptotic constant Cq.
• Derive bounds for Cq and provide small-parameter exact values or constructions for insight.
• Develop skirting arrays as a refinement and connect to covering arrays for bounds.

제안 방법

• Define skirting sets and the total domination framework in G(n,q).
• Use subadditivity and Fekete’s lemma to show f(n,q) = Cq^( (1+o(1))n ).
• Provide lower and upper bounds for Cq via simple counting and constructive arguments.
• Offer explicit constructions for small n,q and compute f(n,q) bounds with ILP/ proofs.
• Introduce skirting arrays and relate SAN(t,n,q) to CAN(t,n,q) to derive bounds.

실험 결과

연구 질문

• RQ1 What is the asymptotic growth rate of f(n,q) as n grows for fixed q?
• RQ2 What bounds can be established on the constant Cq in f(n,q) = Cq^( (1+o(1))n )?
• RQ3 How do small-parameter values (n<q) behave for f(n,q) and what constructions achieve them?
• RQ4 Can we systematically extend skirting sets to skirting arrays and connect to covering arrays for bounds?

주요 결과

• For every q ≥ 2, f(n,q) = Cq^( (1+o(1))n ) with Cq = infn √[n]{f(n,q)} and the sequence Cq is non-increasing in q.
• Lower bound: Cq ≥ q/(q−1) and upper bound: Cq ≤ q^(1/(q−1)) with Cq = 1+ (ln q)/(q−1) + o(1/q).
• When n<q, f(n,q) = n+1 (exact).
• A concrete construction shows f(n,3) and f(n,4) values for small n; f(q,q) ≤ 2q−1 (small parameter bound).
• Skirting arrays SA(N;t,n,q) generalize the concept and relate SAN(t,n,q) to CAN(t,n,q), enabling new bounds like f(q,q) ≤ CAN(t,q,v) − (q−1)^t.
• A practical SA example yields f(20,20) ≤ 35 via an explicit SA(19;4,20,4).