Continued fraction algorithm for Sturmian colorings of trees

TL;DR

This paper introduces an induction algorithm for Sturmian colorings of regular trees, characterizing minimal unbounded factor complexity colorings through a process similar to Sturmian words.

Contribution

It develops a novel induction algorithm for Sturmian colorings of trees and characterizes these colorings using this algorithm.

Findings

Established an induction algorithm for Sturmian colorings.

Characterized Sturmian colorings via the induction data.

Linked tree colorings to Sturmian word properties.

Abstract

Factor complexity $b_\phi(n)$ for a vertex coloring $\phi$ of a regular tree is the number of colored $n$-balls up to color-preserving automorphisms. Sturmian colorings are colorings of minimal unbounded factor complexity $b_\phi(n) = n+2$. In this article, we prove an induction algorithm for Sturmian colorings using colored balls in a way analogous to induction algorithm of Sturmian words. Furthermore, we characterize Sturmian colorings in terms of the data for the induction algorithm.