Perfect 2‐colorings of Hamming graphs

Abstract

We consider the problem of existence of perfect 2‐colorings (equitable 2‐partitions) of Hamming graphs with given parameters. We start with conditions on parameters of graphs and colorings that are necessary for their existence. Next we observe known constructions of perfect colorings and propose some new ones giving new parameters. At last, we deduce which parameters of colorings are covered by these constructions and give tables of admissible parameters of 2‐colorings in Hamming graphs H ( n , q ) for small n and q. Using the connection with perfect colorings, we construct an orthogonal array OA(2048, 7, 4, 5).

Minimum supports of eigenfunctions of graphs: a survey

“…The matrix S = (S i,j ) i,j∈{1,...,r} is called the quotient matrix of the equitable partition. A set C ⊆ V is called a 1-perfect code in G if every ball of radius 1 contains one vertex from C. For more information on equitable partitions and perfect codes we refer the reader to [9], [43,Chapter 5] and [1,46,86,87].…”

Subgroup regular sets in Cayley graphs

“…Perfect 2-colourings are closely related to coding theory and as such have been studied extensively over many years. See, for example, [12] for a study of perfect 2-colorings of Johnson graphs J(v, 3), and [3,23] for some recent results on perfect 2-colourings of Hamming graphs. In [1], equitable partitions of Latin square graphs are studied and those whose quotient matrix does not have an eigenvalue −3 are classified.…”

On the non-existence of extended perfect codes and some perfect colorings

“…These codes originally were defined by Delsarte [18], but here we use the different equivalent definition from [19]. For more information about completely regular codes and problem of its existence, we refer to the survey [20], papers [21,22] (for codes with covering radius ρ = 1), and the small-value tables of parameters [23].…”

“…In addition, we prove that extended 1-perfect codes can exist in H(n, q) only if n is even, which particularly implies the non-existence of some MDS codes with distance 4. We hope that this method can be applied for a proof of the non-existence of some other perfect k-colorings (but for perfect 2-colorings it does not add something new to results from [22]).…”