A graph G(V, E) is word-representable, if there exists a word w over the alphabet V such that for any two distinct letters x and y, \(\{x,y\}\in V\) , x and y alternate in w if and only if \(xy \in E\) . In this paper, we prove that every non-empty word-representable graph can be represented by a word containing no non-trivial squares. This result provides a positive answer to the open problem present in the book Words and graphs written by Sergey Kitaev, and Vadim Lozin. We further prove that, for a word-representable graph G with representation number k, every k-uniform word representing G is also square-free. We also prove that every minimal-length word representing a graph is square-free. Moreover, we count the number of possible square-free word-representations of a complete graph. Then, we provide an example of a non-complete word-representable graph which has a finite number of square-free word-representations. Finally, using the infinite square-free string generated from the Thue-Morse sequence, we prove that there exist infinitely many square-free word-representants for these remaining non-complete connected word-representable graphs.
