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PreprintCC BY-NC-SA 4.0arXiv

An improved lower bound for maximal partial Room squares

  • Peter Horak
  • Dongryul Kim
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Abstract

We establish a new asymptotic lower bound for the volume of a maximal partial Room square of even order n. We prove that its volume is at least (2√(66)−6)/57n2−O(n), improving the leading coefficient 1/8 in the lower bound obtained by Meszka and Rosa. The proof combines two counting arguments, one based on a transversal of empty cells and the other on the graph of occurring symbol pairs.

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Cite

BibTeXRIS
BibTeX
@misc{horak2026improved,
  title = {{An improved lower bound for maximal partial Room squares}},
  author = {Peter Horak and Dongryul Kim},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.07117},
  eprint = {2611.07117},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.07117},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - An improved lower bound for maximal partial Room squares
AU  - Peter Horak
AU  - Dongryul Kim
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.07117
UR  - https://arxiv.org/abs/2611.07117
AB  - We establish a new asymptotic lower bound for the volume of a maximal partial Room square of even order n. We prove that its volume is at least (2√(66)−6)/57n^2−O(n), improving the leading coefficient 1/8 in the lower bound obtained by Meszka and Rosa. The proof combines two counting arguments, one based on a transversal of empty cells and the other on the graph of occurring symbol pairs.
ER  -