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 semi-transitive orientation is an acyclic directed graph where for any directed path $v_0\to v_1\to \cdots \to v_m$, $m\ge 2$ 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 $n,a_1,a_2,\dots ,a_k$, we consider the undirected circulant graph with set of vertices $\{0,1,2,\dots ,n-1\}$ and the set of edges $\{ij\mid |i-j|(modn)$ is in $\{a_1,a_2,\dots ,a_k\}\}$, where $0<a_1<a_2<\cdots <a_k<(n+1)/2$. 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 $\{a_1,a_2,\dots ,a_k\}$ are consecutive positive integers.

This paper examines the semi-transitive orientability of circulant graphs and shows that certain circulant graphs are semi-transitive while others are not, given specific assumptions. Additionally, we provide an upper bound on the representation number of certain $k$-regular circulant graphs.