无向De Bruijn图中$t$ -识别码的存在性
On the Existence of $t$-Identifying Codes in Undirected De Bruijn Graphs
摘要：
本文证明了字符串长度为$n$，字母表长度为$d$的无向de Bruijn图类上$t$ -识别码的存在性，简称为$\mathcal{B}(d,n)$ .结果表明，当$d \geq 3$、$n \geq 2t$和$t \geq 1$时，$\mathcal{B}(d,n)$是$t$可识别的。我们还表明，如果是$d \geq 3$、$n \geq 3$和$t=2$，或者如果是$d = 2$、$n \geq 3$和$t=1$，则$\mathcal{B}(d,n)$是$t$可识别的。其余案件仍在审理中。此外，我们还证明了无向非二元de Bruijn图的偏心率是$n$。
英文摘要：
This paper proves the existence of $t$-identifying codes on the class of undirected de Bruijn graphs with string length $n$ and alphabet size $d$, referred to as $\mathcal{B}(d,n)$. It is shown that $\mathcal{B}(d,n)$ is $t$-identifiable whenever $d \geq 3$ and $n \geq 2t$, and $t \geq 1$. We also show that $\mathcal{B}(d,n)$ is $t$-identifiable if either $d \geq 3$, $n \geq 3$, and $t=2$, or if $d = 2$, $n \geq 3$, and $t=1$. The remaining cases remain open. Additionally, we show that the eccentricity of the undirected non-binary de Bruijn graph is $n$.
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