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A counterexample to Nagamochi's scoring lemma and a new rectangle packing bound — arXiv:2609.37410

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.37410
License (as archived): CC0-1.0
Archived: 2026-10-07T18:33:41.060Z
SHA-256 of the archived text: c2620ff630c92ef89bbc59b6edcd8eccb15dcf2ad9373a2aa43dc0da06e81b3a
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How to cite

A counterexample to Nagamochi's scoring lemma and a new rectangle packing bound — arXiv:2609.37410 — arXiv (en), https://arxiv.org/abs/2609.37410 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_5faacc296cd7 (Metraipolis archive dc_5faacc296cd7, sha256 c2620ff630c92ef89bbc59b6edcd8eccb15dcf2ad9373a2aa43dc0da06e81b3a)

BibTeX

@misc{dc5faacc296cd7,
  title = {A counterexample to Nagamochi's scoring lemma and a new rectangle packing bound — arXiv:2609.37410},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.37410},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_5faacc296cd7; sha256 c2620ff630c92ef89bbc59b6edcd8eccb15dcf2ad9373a2aa43dc0da06e81b3a},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.37410. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Let $s(N)$ denote the smallest side length of a square containing $N$ unit squares with arbitrary orientations and pairwise disjoint interiors. Nagamochi's Packing Unit Squares in a Rectangle (2005) states a rectangle packing bound from which he deduces two infinite families of exact values: $s(k^2-1) = k$ and $s(k^2-2) = k$ for every integer $k \geq 2$. We construct a family of counterexamples, local to a corner of the container, to the scoring assertion in Nagamochi's Lemma 1. These counterexamples show that the published proof of the rectangle bound is incomplete, but do not disprove the bound itself. We then give an independent proof of a weaker rectangle bound using a strip measure. This recovers $s(k^2-1) = k$ for every integer $k \geq 2$ and yields an explicit lower bound for $s(N)$ that improves strictly on the area bound for every nonsquare integer $N \geq 8$. Our argument does not establish Nagamochi's full rectangle bound or the identity $s(k^2-2) = k$.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-30 in math.CO).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.37410 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-07T17:59:19.225Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-30?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).