On word-representability of simplified de Bruijn graphs

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy\in E$. Word-representable graphs generalize several important classes of graphs such as $3$-colorable graphs, circle graphs, and comparability graphs. There is a long line of research in the literature dedicated to word-representable graphs. In this paper, we study word-representability of simplified de Bruijn graphs. The simplified de Bruijn graph $S(n,k)$ is a simple graph obtained from the de Bruijn graph $B(n,k)$ by removing orientations and loops and replacing multiple edges between a pair of vertices by a single edge. De Bruijn graphs are a key object in combinatorics on words that found numerous applications, in particular, in genome assembly. We show that binary simplified de Bruijn graphs (i.e.\ $S(n,2)$) are word-representable for any $n\geq 1$, while $S(2,k)$ and $S(3,k)$ are non-word-representable for $k\geq 3$. We conjecture that all simplified de Bruijn graphs $S(n,k)$ are non-word-rerpesentable for $n\geq 4$ and $k\geq 3$.