On semi-transitive orientability of 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 $w$. 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.