A note on the paper “On Brlek-Reutenauer conjecture”

Abstract

In this short note we point to an error in the proof of a theorem stated in [L. Balkov\'a \& E. Pelantov\'a \& \v S. Starosta, On Brlek-Reutenauer conjecture, \emph{Theoret. Comput. Sci.} {\bf 412} (2011), 5649--5655]. By constructing a counterexample, we show that the assertion of the theorem is actually incorrect. Although this theorem is of a technical character, it was used in an argument leading to a corollary of a general interest to the Brlek-Reutenauer conjecture, and thus as a consequence of this note we have that the proof of the mentioned corollary is also flawed.