A 1-11-representation of a graph $G(V,E)$ is a word over the alphabet $V$ such that two distinct vertices $x$ and $y$ are adjacent if and only if the restricted word $w{x,y}$ (obtained from $w$ by deleting all letters except $x$ and $y$) contains at most one occurrence of $xx$ or $yy$. Although every graph admits a 1-11-representation, the repetition patterns that may or must appear in such representations have not been fully studied. In this paper, we study cube-free and square-free 1-11-representations of graphs. We first show that cubes cannot always be avoided in 1-11-representations of minimum length by providing a graph for which every minimum-length 1-11-representation necessarily contains a cube. We then focus on permutational 1-11-representations, where the representing word is a concatenation of permutations of the vertex set. In this setting, we prove that any cube appearing in a permutational 1-11-representation can be removed without changing the represented graph. As a consequence, every permutational 1-11-representation attaining the permutational 1-11-representation number is cube-free. We further show that this behaviour does not extend to squares by providing a graph for which every permutational 1-11-representation with the minimum number of permutations necessarily contains a square. Finally, we prove that the language of all 1-11-representations of a given graph is regular. Moreover, we show that the language of all permutational 1-11-representations of a graph is also regular.

翻译：图$G(V,E)$的1-11表示是指字母表$V$上的一个字$w$，使得两个不同顶点$x$和$y$相邻当且仅当限制字$w_{\{x,y\}}$（通过从$w$中删除除$x$和$y$外的所有字母得到）至多出现一次$xx$或$yy$。尽管每个图都允许1-11表示，但此类表示中可能出现或必须出现的重复模式尚未得到充分研究。本文研究图的立方自由和平方自由1-11表示。我们首先证明，通过构造一个图，其所有最小长度的1-11表示必然包含一个立方，从而表明在最小长度的1-11表示中立方并非总能避免。随后我们关注置换1-11表示，其中表示字是顶点集置换的串联。在此设定下，我们证明出现在置换1-11表示中的任何立方都可以在不改变所表示图的情况下被移除。因此，达到置换1-11表示数的每个置换1-11表示都是立方自由的。我们进一步证明，此性质不能推广到平方，通过构造一个图，其所有具有最少置换数量的置换1-11表示必然包含一个平方。最后，我们证明给定图的所有1-11表示的语言是正则的。此外，我们还证明图的所有置换1-11表示的语言也是正则的。