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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| Vydané v: | Graphs and combinatorics Ročník 34; číslo 5; s. 947 - 964 |
|---|---|
| Hlavní autori: | , , |
| Médium: | Journal Article Publikácia |
| Jazyk: | English |
| Vydavateľské údaje: |
Tokyo
Springer Japan
01.09.2018
Springer Nature B.V |
| Predmet: | |
| ISSN: | 0911-0119, 1435-5914 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | 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
. |
|---|---|
| Bibliografia: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0911-0119 1435-5914 |
| DOI: | 10.1007/s00373-018-1923-7 |