Distance-2 MDS Codes and Latin Colorings in the Doob Graphs

The maximum independent sets in the Doob graphs D ( m ,  n ) are analogs of the distance-2 MDS codes in Hamming graphs and of the latin hypercubes. We prove the characterization of these sets stating that every such set is semilinear or reducible. As related objects, we study vertex sets with maximu...

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Bibliographic Details
Published inGraphs and combinatorics Vol. 34; no. 5; pp. 1001 - 1017
Main Authors Krotov, Denis S., Bespalov, Evgeny A.
Format Journal Article
LanguageEnglish
Published Tokyo Springer Japan 01.09.2018
Springer Nature B.V
Subjects
Combinatorics
Eigenvalues
Engineering Design
Graphs
Hypercubes
Mathematics
Mathematics and Statistics
Original Paper
Vertex sets
Equitable partition
94B25
Doob graph
MDS code
Completely regular set
Maximum independent set
Maximum cut
Latin hypercube
05B15
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Summary:The maximum independent sets in the Doob graphs D ( m ,  n ) are analogs of the distance-2 MDS codes in Hamming graphs and of the latin hypercubes. We prove the characterization of these sets stating that every such set is semilinear or reducible. As related objects, we study vertex sets with maximum cut (edge boundary) in D ( m ,  n ) and prove some facts on their structure. We show that the considered two classes (the maximum independent sets and the maximum-cut sets) can be defined as classes of completely regular sets with specified 2-by-2 quotient matrices. It is notable that for a set from the considered classes, the eigenvalues of the quotient matrix are the maximum and the minimum eigenvalues of the graph. For D ( m , 0), we show the existence of a third, intermediate, class of completely regular sets with the same property.
ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-018-1926-4