Squarefree word

Squarefree word

A squarefree word is a word that does not contain any subword twice in a row. There exist infinite squarefree words in any alphabet with three or more symbols, as proved by Axel Thue. To build an infinite squarefree word in the alphabet {"a, b, c"}, start with any word w_1 from this alphabet and obtain the word w_{i+1} by replacing each "a" in w_1 with "abcbacbcabcba", each "b" with "bcacbacabcacb", and each "c" with "cabacbabcabac" (this example was shown by J. Leech). It is possible to check that the sequence converges to the infinite squarefree word "abcbacbcabcbabcacbacabcacbcabacbabcabacbcacbacabcacb"...

Over a two-letter alphabet {"a, b"} the only square-free words are the empty word and "a", "b", "ab", "ba", "aba", and "bab".

The Thue number of a graph "G" is the smallest number "k" such that "G" has a "k"-coloring for which the sequence of colors along every non-repeating path is squarefree.


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