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A Classification of Hyperfocused 12-Arcs

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Abstract

A k-arc in PG(2, q) is a set of k points no three of which are collinear. A hyperfocused k-arc is a k-arc in which the \(k \atopwithdelims ()2\) secants meet some external line in exactly \(k-1\) points. Hyperfocused k-arcs can be viewed as 1-factorizations of the complete graph \(K_k\) that embed in PG(2, q). We study the 526,915,620 1-factorizations of \(K_{12}\), determine which are embeddable in PG(2, q), and classify hyperfocused 12-arcs. Specifically we show that if a 12-arc \(\mathcal {K}\) is a hyperfocused arc in PG(2, q) then \(q = 2^{5k}\) and \(\mathcal {K}\) is a subset of a hyperconic including the nucleus.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the authors on reasonable request.

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Acknowledgements

This research started at the Rocky Mountain–Great Plains Graduate Research Workshop in Combinatorics that was held at the University of Colorado Denver and the University of Denver in summer 2014, and we thank them for their hospitality. We thank Petteri Kaski and Patric Östergård for providing us with the data that allowed us to complete this classification. We also thank William Cherowitzo for his many helpful remarks.

Funding

Research supported in part by U.S. National Science Foundation grant DMS-1427526 for The Rocky Mountain–Great Plains Graduate Research Workshop in Combinatorics, and Collaboration Grants from the Simons Foundation, #316262 to Stephen G. Hartke, #711898 to Jason Williford.

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Authors and Affiliations

  1. Department of Mathematics, Westfield State University, 577 Western Ave, Westfield, MA, 01086, USA

    Philip DeOrsey

  2. Department of Mathematical and Statistical Sciences, University of Colorado Denver, Campus Box 170, PO Box 173364, Denver, CO, 80217-3364, USA

    Stephen G. Hartke

  3. Deparment of Mathematics and Statistics, University of Wyoming, Dept 3036, 1000 E University Ave, Laramie, WY, 82071, USA

    Jason Williford

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  1. Philip DeOrsey
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  2. Stephen G. Hartke
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  3. Jason Williford
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Correspondence to Jason Williford.

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Research supported in part by U.S. National Science Foundation grant DMS-1427526 for The Rocky Mountain–Great Plains Graduate Research Workshop in Combinatorics, and Collaboration Grants from the Simons Foundation (#316262 to Stephen G. Hartke, #711898 to Jason Williford).

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DeOrsey, P., Hartke, S.G. & Williford, J. A Classification of Hyperfocused 12-Arcs. Graphs and Combinatorics 38, 151 (2022). https://doi.org/10.1007/s00373-022-02547-2

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