焦饼图中的圈
Cycles in the burnt pancake graphs
摘要：
饼图$P_n$是对称群$S_n$在由前缀反转生成的$n$元素上的Cayley图。$P_n$已被证明具有使其成为并行处理器的有用网络方案的属性。例如，它是$(n-1)$ -正则的，顶点可传递的，并且可以用$6\leq\ell\leq n!$在其中嵌入长度为$\ell$的循环。焦饼图$BP_n$是使用前缀反转作为生成元的带符号置换群$B_n$的凯莱图，具有类似的性质。实际上，$BP_n$是$n$ -正则的和顶点传递的。在本文中，我们证明了$BP_n$有每个周期长度$\ell$与$8\leq\ell\leq 2^n n!$。所给的证明是利用$BP_n$的递归结构的构造性证明。我们还提出了一个完整的表征$BP_n$中的所有$8$ -周期的$n \geq 2$，这是最小的周期嵌入在$BP_n$，通过呈现其规范形式的产品的前缀反转生成器。
英文摘要：
The pancake graph $P_n$ is the Cayley graph of the symmetric group $S_n$ on $n$ elements generated by prefix reversals. $P_n$ has been shown to have properties that makes it a useful network scheme for parallel processors. For example, it is $(n-1)$-regular, vertex-transitive, and one can embed cycles in it of length $\ell$ with $6\leq\ell\leq n!$. The burnt pancake graph $BP_n$, which is the Cayley graph of the group of signed permutations $B_n$ using prefix reversals as generators, has similar properties. Indeed, $BP_n$ is $n$-regular and vertex-transitive. In this paper, we show that $BP_n$ has every cycle of length $\ell$ with $8\leq\ell\leq 2^n n!$. The proof given is a constructive one that utilizes the recursive structure of $BP_n$. We also present a complete characterization of all the $8$-cycles in $BP_n$ for $n \geq 2$, which are the smallest cycles embeddable in $BP_n$, by presenting their canonical forms as products of the prefix reversal generators.
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