To the theory of \(q\)-ary Steiner and other-type trades

We introduce the concept of a clique bitrade, which generalizes several known types of bitrades, including latin bitrades, Steiner \(T(k-1,k,v)\) bitrades, extended \(1\)-perfect bitrades. For a distance-regular graph, we show a one-to-one correspondence between the clique bitrades that meet the wei...

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Veröffentlicht in:arXiv.org 2015-08
Hauptverfasser: Krotov, Denis, Mogilnykh, Ivan, Potapov, Vladimir
Format: Artikel
Sprache:eng
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Zusammenfassung:We introduce the concept of a clique bitrade, which generalizes several known types of bitrades, including latin bitrades, Steiner \(T(k-1,k,v)\) bitrades, extended \(1\)-perfect bitrades. For a distance-regular graph, we show a one-to-one correspondence between the clique bitrades that meet the weight-distribution lower bound on the cardinality and the bipartite isometric subgraphs that are distance-regular with certain parameters. As an application of the results, we find the minimum cardinality of \(q\)-ary Steiner \(T_q(k-1,k,v)\) bitrades and show a connection of minimum such bitrades with dual polar subgraphs of the Grassmann graph \(J_q(v,k)\). Keywords: bitrades, trades, Steiner systems, subspace designs
ISSN:2331-8422
DOI:10.48550/arxiv.1412.3792