## Electronic Communications in Probability

### Symmetric 1-dependent colorings of the integers

#### Abstract

In a recent paper, we constructed a stationary $1-$dependent $4-$coloring of the integers that is invariant under permutations of the colors. This was the first stationary $k-$dependent $q-$coloring for any $k$ and $q$. When the analogous construction is carried out for $q>4$ colors, the resulting process is not $k-$dependent for any $k$. We construct here a process that is symmetric in the colors and $1-$dependent for every $q\geq 4$. The construction uses a recursion involving Chebyshev polynomials evaluated at $\sqrt{q}/2$.

#### Article information

Source
Electron. Commun. Probab., Volume 20 (2015), paper no. 31, 8 pp.

Dates
Accepted: 29 March 2015
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465320958

Digital Object Identifier
doi:10.1214/ECP.v20-4070

Mathematical Reviews number (MathSciNet)
MR3327870

Zentralblatt MATH identifier
06473030

Subjects
Primary: 60G10: Stationary processes
Secondary: 05C15: Coloring of graphs and hypergraphs 60C05: Combinatorial probability

Rights