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Graph Divisible Designs and Packing Constructions

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Abstract

We introduce a generalization of group divisible designs and offer example applications to challenging problems in design theory. The generalization considers edge-decompositions of joins of arbitrary graphs, whereas group divisible designs handle only joins of edgeless graphs. Our example constructions include: (1) optimal packings with block size five for the previously unsettled congruence class \(v \equiv 13 \pmod {20}\); (2) an optimal grooming with with ratio seven for the previously unsettled congruence class \(v \equiv 56 \pmod {84}\); and (3) a constructive ‘quadratic’ embedding of partial designs with block size four.

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Authors and Affiliations

  1. Mathematics and Statistics, University of Victoria, Victoria, BC, Canada

    Peter J. Dukes

  2. Computer Science, University of Vermont, Burlington, VT, USA

    Alan C. H. Ling

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  1. Peter J. Dukes
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  2. Alan C. H. Ling
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Correspondence to Peter J. Dukes.

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Research of Peter Dukes is supported by NSERC.

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Dukes, P.J., Ling, A.C.H. Graph Divisible Designs and Packing Constructions. Graphs and Combinatorics 31, 2181–2191 (2015). https://doi.org/10.1007/s00373-014-1518-x

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  • DOI: https://doi.org/10.1007/s00373-014-1518-x

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