On a Problem by Shapozenko on Johnson Graphs

The Johnson graph J ( n ,  m ) has the m -subsets of { 1 , 2 , … , n } as vertices and two subsets are adjacent in the graph if they share m - 1 elements. Shapozenko asked about the isoperimetric function μ n , m ( k ) of Johnson graphs, that is, the cardinality of the smallest boundary of sets with...

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Bibliographic Details
Published in:Graphs and combinatorics Vol. 34; no. 5; pp. 947 - 964
Main Authors: Diego, Víctor, Serra, Oriol, Vena, Lluís
Format: Journal Article Publication
Language:English
Published: Tokyo Springer Japan 01.09.2018
Springer Nature B.V
Subjects:
05 Combinatorics
05C Graph theory
Anàlisi numèrica
Boolean algebra
Classificació AMS
Combinatorics
Engineering Design
Grafs, Teoria de
Graph theory
Graphs
Isoperimetric problem
Johnson graph
Matemàtiques i estadística
Mathematics
Mathematics and Statistics
Original Paper
Shift compression
Upper bounds
Àrees temàtiques de la UPC
Isoperimetric problem
Johnson graph
Shift compression
ISSN:0911-0119, 1435-5914
Online Access:Get full text
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Summary:The Johnson graph J ( n ,  m ) has the m -subsets of { 1 , 2 , … , n } as vertices and two subsets are adjacent in the graph if they share m - 1 elements. Shapozenko asked about the isoperimetric function μ 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 ≤ k ≤ n m . We give an upper bound for μ 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 ≤ m + 1 and for all values of k when m = 2 .
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ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-018-1923-7