An alternating colouring function on strings

TL;DR

This paper introduces an alternating colouring function on binary strings, characterizes non-colourable strings, and relates their count to Jacobsthal numbers, revealing structural properties of associated subshifts and de Bruijn graphs.

Contribution

It defines a novel alternating colouring function on strings, establishes a formula for non-colourable strings using Jacobsthal numbers, and analyzes the structure of de Bruijn graphs with non-colourable edges removed.

Findings

Number of non-colourable strings of length n is 2*(J_{n-2}+1).

Points in the subshift alternate between two colours.

Number of sources and sinks in the modified de Bruijn graph is K_n - 4.

Abstract

An alternating colouring function is defined on strings over the alphabet $\{0, 1\}$. It divides the strings in colourable and non-colourable ones. The points in the subshift of finite type defined by forbidding all non-colourable strings of a certain length alternate between states of one colour and states of the other colour. In other words, the points in the 2nd power shifts all have the same colour. The number $K_n$ of non-colourable strings of length $n \ge 2$ is shown to be $2 \cdot (J_{n-2} + 1)$ where $J$ is the sequence of Jacobsthal numbers. The number of sources and sinks in the de Bruijn graph of dimension $n \ge 3$ with non-colourable edges removed is shown each to be $K_n - 4$.