The perfect $2$-colorings of infinite circulant graphs with a continuous set of odd distances

Abstract

A vertex coloring of a given simple graph G = (V, E) with k colors (k-coloring) is a map from its vertex set to the set of integers {1, 2, 3, . . . , k}. A coloring is called perfect if the multiset of colors appearing on the neighbours of any vertex depends only on the color of the vertex. We consider perfect colorings of Cayley graphs of the additive group of integers with generating set {1, −1, 3, −3, 5, −5, . . . , 2n− 1, 1 − 2n} for a positive integer n. We enumerate perfect 2-colorings of the graphs under consideration and state the conjecture generalizing the main result to an arbitrary number of colors.

Polynomial method for perfect 2-colourings of circulant graphs

“…On the contrary, in this work we prove some inequalities between permissible values of b, c, k, which apply to arbitrary values of l 1 , ..., l k . In particular, we prove the hypothesis (stated in [1]) that the parameters (5,3) are not permissible for 3 distances.…”

“…3) No infinite circulant graph with 4 distances has a perfect 2-colouring with parameters (6, 5), (7,4), (8,3), (7,5), (7,6), (8,5), (8,6) or (8,7).…”

“…Perfect 2-colourings of circulant graphs and their parameters are being subject of active research (see, e.g., [1], [2], [3], [4], [5]). However, the above works consider only the cases when the distances l 1 , ..., l k have some special form.…”

On a metric property of perfect colorings

“…Our results can be widely used for perfect colorings in circulant graphs. Perfect colorings in some classes of such graphs were previously studied in [4,5].…”