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A graph $G = (V, E)$ is said to be word-representable if a word $w$ can be formed using the letters of the alphabet $V$ such that for every pair of vertices $x$ and $y$, $xy \in E$ if and only if $x$ and $y$ alternate in $w$. A \textit{semi-transitive} orientation is an acyclic directed graph where for any directed path $v_0 \rightarrow v_1 \rightarrow \ldots \rightarrow v_m$, $m \ge 2$ either there is no arc between $v_0$ and $v_m$ or for all $1 \le i < j \le m$ there is an arc between $v_i$ and $v_j$. An undirected graph is semi-transitive if it admits a semi-transitive orientation. For given positive integers $n, a_1, a_2, \ldots, a_k$, we consider the undirected circulant graph with set of vertices $\{0, 1, 2, \ldots, n-1\}$ and the set of edges$\{ij ~ | ~ (i - j) \pmod n$ or $(j-i) \pmod n$ are in $\{a_1, a_2, \ldots, a_k\}\}$, where $ 0 < a_1 < a_2 < \ldots < a_k < (n+1)/2$. Recently, Kitaev and Pyatkin have shown that every $4$-regular circulant graph is semi-transitive. Further, they have posed an open problem regarding the semi-transitive orientability of circulant graphs for which the elements of the set $\{a_1, a_2, \ldots, a_k\}$ are consecutive positive integers. In this paper, we solve the problem mentioned above. In addition, we show that under certain assumptions, some $k(\ge5)$-regular circulant graphs are semi-transitive, and some are not. Moreover, since a semi-transitive orientation is a characterisation of word-representability, we give some upper bound for the representation number of certain $k$-regular circulant graphs.

A graph $G = (V, E)$ is said to be word-representable if a word $w$ can be formed using the letters of the alphabet $V$ such that for every pair of vertices $x$ and $y$, $xy \in E$ if and only if $x$ and $y$ alternate in $w$. Gaetz and Ji have recently introduced the notion of minimum length word-representants for word-representable graphs. They have also determined the minimum possible length of the word-representants for certain classes of graphs, such as trees and cycles. It is know that Cartesian and Rooted products preserve word-representability. Moreover, Broere constructed a uniform word representing the Cartesian product of $G$ and $K_n$ using occurrence based functions. In this paper, we study the minimum length of word-representants for Cartesian and Rooted products using morphism and occurrence based function, respectively. Also, we solve an open problem posed by Broere in his master thesis. This problem asks to construct a word for the Cartesian product of two arbitrary word-representable graphs., Comment: 14 pages, submitted to DAM

A graph $G = (V, E)$ is word-representable, if there exists a word w over the alphabet V such that for letters ${x, y} \in V$ , $x$ and $y$ alternate in $w$ if and only if $xy \in E$. In this paper, we prove that any 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. Also, we prove that for a word-representable graph $G$, if the representation number of $G$ is $k$, then every $k$-uniform word representing the graph $G$ is also square-free. Moreover, we prove that every minimal-length word representing a graph is square-free. Then, we count the number of possible square-free word-representations of a complete graph. At last, using the infinite square-free string generated from the Thue-Morse sequence, we prove that infinitely many square-free words represent a non-complete connected word-representable graph., Comment: 12 pages