Abstract
The Johnson graph J(n, m) has the m-subsets of \(\{1,2,\ldots ,n\}\) as vertices and two subsets are adjacent in the graph if they share \(m-1\) elements. Shapozenko asked about the isoperimetric function \(\mu _{n,m}(k)\) of Johnson graphs, that is, the cardinality of the smallest boundary of sets with k vertices in J(n, m) for each \(1\le k\le {n\atopwithdelims ()m}\). We give an upper bound for \(\mu _{n,m}(k)\) and show that, for each given k such that the solution to the Shadow Minimization Problem in the Boolean lattice is unique, and each sufficiently large n, the given upper bound is tight. We also show that the bound is tight for the small values of \(k\le m+1\) and for all values of k when \(m=2\).
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Acknowledgements
The authors are grateful to the comments and remarks of the referees, pointing out some inaccuracies in the original manuscript and helping to improve the readability of the paper. V. Diego acknowledges support from Ministerio de Ciencia e Innovación, Spain, and the European Regional Development Fund under a FPI Grant in the project MTM2011-28800-C02-01. O. Serra was supported by the Spanish Ministerio de Economía y Competitividad under project MTM2014-54745-P. L. Vena was supported by the Center of Excellence-Inst. for Theor. Comp. Sci., Prague, P202/12/G061, by Project ERCCZ LL1201 CORES, and by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 339109.
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Diego, V., Serra, O. & Vena, L. On a Problem by Shapozenko on Johnson Graphs. Graphs and Combinatorics 34, 947–964 (2018). https://doi.org/10.1007/s00373-018-1923-7
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DOI: https://doi.org/10.1007/s00373-018-1923-7