On the Word-Representability of 5-Regular Circulant Graphs

A graph $G = (V, E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that, for any two distinct vertices $x, y \in V$, $xy \in E$ if and only if $x$ and $y$ alternate in $w$. Two letters $x$ and $y$ are said to alternate in $w$ if, after removing all other letters from $w$, the resulting word is of the form $xyxy\dots$ or $yxyx\dots$ (of even or odd length). For a given set $R = \{r_1, r_2, \dots, r_k\}$ of jump elements, an undirected circulant graph $C_n(R)$ on $n$ vertices has vertex set $\{0, 1, \dots, n-1\}$ and edge set $ E = \left\{ \{i,j\} \;\middle|\; |i - j| \bmod n \in \{r_1, r_2, \dots, r_k\} \right\}, $ where $0 < r_1 < r_2 < \dots < r_k < \frac{n}{2}$. Recently, Kitaev and Pyatkin proved that every 4-regular circulant graph is word-representable. Srinivasan and Hariharasubramanian further investigated circulant graphs and obtained bounds on the representation number for $k$-regular circulant graphs with $2 \le k \le 4$. In addition to these positive results, their work also presents examples of non-word-representable circulant graphs. In this work, we study word-representability and the representation number of 5-regular circulant graphs via techniques from elementary number theory and group theory, as well as graph coloring, graph factorization and morphisms.