Distance regular colorings of an infinite rectangular grid

Abstract

We study the infinite graph of n-dimensional rectangular grid that doesn't appear distance regular and the distance regular colorings of this graph, which are defined as the distance colorings with respect to completely regular codes. It is proved that the elements of the parameter matrix of an arbitrary distance regular coloring form two monotonic sequences. It is shown that every irreducible distance regular coloring of the n-dimensional rectangular grid has at most 2n + 1 colors.

On perfect colorings of infinite multipath graphs

“…Describe all reduced perfect colorings of the graph under consideration to which belong that semicoloring. It is easy to see that these are the colorings S(1), S (2), and S 11 (3). Note that S 11 (3) cannot be obtained by gluing the colors of the perfect coloring: the corresponding 4-periodic perfect colorings get glued immediately in S (2).…”

“…In this case the sets of vertices of the first and last colors form complete regular codes. The parameters of all distance regular colorings of the infinite rectangular grid were listed by Avgustinovich, Vasil' eva, and Sergeeva in [3].…”

Perfect colorings of the infinite square grid: coverings and twin colors

“…Next, we consider the case where two nodes with difference in {[±3, ±1], [±1, ±3]} (w.l.o.g., [3,1]) have the same color. In both cases, A is an orbit coloring.…”

“…Perfect colorings of G(Z 2 ) were studied in [2,5,3] (list of colorings up to 9 colors), [4] (results about periodicity), [1] (characterization of completely regular codes).…”

“…In this paper, we (1) to characterize coverings by G(Z 2 ) in the case when the target graph is bipartite, and (2) classify all bipartite perfect colorings of G(Z 2 ) with twin colors.…”