Subword complexity and Sturmian colorings of regular trees

Abstract

Abstract. In this article, we study subword complexity of colorings of regular trees. We characterize colorings of bounded subword complexity and study Sturmian colorings, which are colorings of minimal unbounded subword complexity.We classify Sturmian colorings using their type sets. We show that any Sturmian coloring is a lifting of a coloring on a quotient graph of the tree which is a geodesic or a ray with loops possibly attached, thus a lifting of an "infinte word". We further give a complete characterization of the quotient graph for eventually periodic ones.

Continued fraction algorithm for Sturmian colorings of trees

“…A Sturmian coloring φ on a tree T is called of bounded type if the type set of each vertex is finite. If one vertex is of bounded type, then all vertices are of bounded type [8].…”

“…Factor complexity was generalized from sequences to vertex colorings of a regular tree in [8] and [7]. For a graph X , let us denote its vertex set by V X and its set of oriented edges by E X .…”

“…For a vertex coloring φ : V T → A of a dregular tree T , let B n (φ) be the set of classes of n-balls appearing in T colored by φ up to color-preserving isomorphisms of n-balls. The factor complexity b n (φ) (called subword complexity in [8]) is the cardinality of the set B n (φ).…”

Colorings of Trees With Linear, Intermediate and Exponential Subball Complexity

“…The subball complexity b φ (n) of a coloring φ, defined by D. Kim and the second author (which is called subword complexity of a coloring in [5]), is the number of colored n-balls in the colored tree (T, φ) up to color-preserving isomorphisms (see the paragraph after Definition 2.2). The subball complexity b φ (n) is closely related to the well-known subword complexity of a sequence, which is an important tool to study sequences which are not periodic.…”

“…Kim and the second author showed a theorem analogous to CovenHedlund Theorem for colorings of regular trees, namely that b φ (n) is bounded if and only if φ is periodic. They further studied Sturmian colorings, which are colorings of minimal unbounded subball complexity b(n) = n + 2 (see [5]). …”