On the bijective colouring of Cantor trees based on transducers

TL;DR

This paper investigates when vertex colourings of infinite n-ary Cantor trees can induce bijective colourings of infinite paths, providing conditions and constructions using Mealy automata, and characterizing cases where finite automata suffice.

Contribution

It establishes necessary and sufficient conditions for bijective colourings of Cantor trees using automata, including an effective construction and limitations for finite automata.

Findings

Bijective colourings exist if and only if n ≥ m.

Effective construction of bijective colourings via Mealy automata.

Finite automata only produce trivial bijective colourings when m=n.

Abstract

Given a vertex colouring of the infinite $n$-ary Cantor tree with $m$ colours ($n,m\geq 2$), the natural problem arises: may this colouring induce a bijective colouring of the infinite paths starting at the root, i.e., that every infinite $m$-coloured string is used for some of these paths but different paths are not coloured identically? In other words, we ask if the above vertex colouring may define a bijective short map between the corresponding Cantor spaces. We show that the answer is positive if and only if $n\geq m$, and provide an effective construction of the bijective colouring in terms of Mealy automata and functions defined by such automata. We also show that a finite Mealy automaton may define such a bijective colouring only in the trivial case, i.e. $m=n$.