On the bijective colouring of Cantor trees based on transducers

Autor: Adam Woryna
Rok vydání: 2022
Předmět:
Zdroj: Discrete Mathematics. 345:112855
ISSN: 0012-365X
DOI: 10.1016/j.disc.2022.112855
Popis: 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. $m=n$.
Databáze: OpenAIRE