Doubled Patterns with Reversal Are 3-Avoidable

In combinatorics on words, a word w over an alphabet \(\varSigma \) is said to avoid a pattern p over an alphabet \(\varDelta \) if there is no factor f of w such that \(f=h(p)\) where \(h:\varDelta ^*\rightarrow \varSigma ^*\) is a non-erasing morphism. A pattern p is said to be k-avoidable if there exists an infinite word over a k-letter alphabet that avoids p. A pattern is doubled if every variable occurs at least twice. Doubled patterns are known to be 3-avoidable. Currie, Mol, and Rampersad have considered a generalized notion which allows variable occurrences to be reversed. That is, \(h(V^R)\) is the mirror image of h(V) for every \(V\in \varDelta \). We show that doubled patterns with reversal are 3-avoidable.