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 $xy\in E$. For integers $n>k>0 $, the shift graph $G(n,k)$ is the graph whose vertex set consists of all increasing $k$-tuples $(x_1,x_2,\dots,x_k)$ with $1\le x_1<x_2<\cdots<x_k\le n$, where two vertices $(x_1,\dots,x_k)$ and $(y_1,\dots,y_k)$ are adjacent whenever $x_{i+1}=y_i$ for all $1\le i\le k-1$ or $y_{i+1}=x_i$ for all $1\le i\le k-1$. Shift graphs are classical examples of sparse graphs having arbitrarily high chromatic number and odd girth. We further observe that shift graphs arise naturally as induced subgraphs of simplified de Bruijn graphs. Although simplified de Bruijn graphs contain non-word-representable members in general, we prove that the entire class of shift graphs is word-representable. We also introduce a natural generalization of shift graphs in which adjacency is defined by more than one shift condition, and show that these generalized shift graphs are likewise word-representable. As a consequence, we obtain an explicit family of graphs exhibiting a contrast between line graph and line digraph constructions: there exists a family of word-representable graphs whose line graphs are not word-representable when the number of vertices is at least $5$, while their line digraphs are word-representable.

翻译：图 $G=(V,E)$ 称为词可表示的，若存在一个字母表 $V$ 上的词 $w$，使得字母 $x$ 和 $y$ 在 $w$ 中交替出现当且仅当 $xy\in E$。对于满足 $n>k>0$ 的整数，移位图 $G(n,k)$ 的顶点集由所有递增 $k$ 元组 $(x_1,x_2,\dots,x_k)$ 构成，其中 $1\le x_1<x_2<\cdots<x_k\le n$，且两个顶点 $(x_1,\dots,x_k)$ 和 $(y_1,\dots,y_k)$ 相邻当且仅当对所有 $1\le i\le k-1$ 有 $x_{i+1}=y_i$，或对所有 $1\le i\le k-1$ 有 $y_{i+1}=x_i$。移位图是具有任意高色数和奇数围长的稀疏图的经典例子。我们进一步观察到，移位图自然作为简化德布鲁因图的导出子图出现。尽管简化德布鲁因图一般包含非词可表示的成员，我们证明整类移位图是词可表示的。我们还引入了移位图的自然推广，其中邻接关系由不止一个移位条件定义，并证明这些广义移位图同样词可表示。作为推论，我们得到了一个显式图族，揭示了线图与线有向图构造之间的对比：存在一族词可表示图，当其顶点数至少为$5$时，其线图不是词可表示的，而其线有向图却是词可表示的。