Distance-2 MDS codes and latin colorings in the Doob graphs

TL;DR

This paper characterizes maximum independent sets in Doob graphs, showing they are either semilinear or reducible, and explores related maximum cut sets and their algebraic properties.

Contribution

It provides a complete characterization of maximum independent sets in Doob graphs and analyzes their algebraic structure and related maximum cut sets.

Findings

Maximum independent sets are semilinear or reducible.

Maximum cut sets are completely regular with specific quotient matrices.

Existence of an intermediate class of regular sets in D(m,0).

Abstract

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.