k方体是k-可表示的
The k-cube is k-representable
摘要：
一个图被称为$k$ -可表示的，如果在图的节点上存在一个词$w$，每个节点正好出现$k$次，使得两个节点之间存在边$x,y$，当且仅当从$w$中删除所有不同于$x,y$的字母后，仍然存在一个词，其中$x,y$交替。证明了如果$G$对$k>1$是$k$ -可表示的，则$G$与$n$节点上的完全图的笛卡尔积是$(k+n-1)$ -可表示的.作为一个直接的结果，$k$ -立方体对于每个$k \geq 1$都是$k$ -可表示的.我们的主要技术包括探索基于出现的函数，该函数将单词$w$中的符号$x$的每个$i$次出现替换为字符串$h(x,i)$。我们构造来实现我们的主要定理的代表词纯粹是由串联和基于出现的函数组成的。
英文摘要：
A graph is called $k$-representable if there exists a word $w$ over the nodes of the graph, each node occurring exactly $k$ times, such that there is an edge between two nodes $x,y$ if and only after removing all letters distinct from $x,y$, from $w$, a word remains in which $x,y$ alternate. We prove that if $G$ is $k$-representable for $k>1$, then the Cartesian product of $G$ and the complete graph on $n$ nodes is $(k+n-1)$-representable. As a direct consequence, the $k$-cube is $k$-representable for every $k \geq 1$. Our main technique consists of exploring occurrence based functions that replace every $i$th occurrence of a symbol $x$ in a word $w$ by a string $h(x,i)$. The representing word we construct to achieve our main theorem is purely composed from concatenation and occurrence based functions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
