We say that a Hamilton cycle $C=(x_1,\ldots,x_n)$ in a graph $G$ is $k$-symmetric, if the mapping $x_i\mapsto x_{i+n/k}$ for all $i=1,\ldots,n$, where indices are considered modulo $n$, is an automorphism of $G$. In other words, if we lay out the vertices $x_1,\ldots,x_n$ equidistantly on a circle and draw the edges of $G$ as straight lines, then the drawing of $G$ has $k$-fold rotational symmetry, i.e., all information about the graph is compressed into a $360^\circ/k$ wedge of the drawing. The maximum $k$ for which there exists a $k$-symmetric Hamilton cycle in $G$ is referred to as the Hamilton compression of $G$. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases we determine their Hamilton compression exactly, and in other cases we provide close lower and upper bounds. The constructed cycles have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced.

翻译：我们称图$G$中的一个哈密顿圈$C=(x_1,\ldots,x_n)$为$k$-对称的，如果映射$x_i\mapsto x_{i+n/k}$（对所有$i=1,\ldots,n$，指标模$n$）是$G$的一个自同构。换言之，若将顶点$x_1,\ldots,x_n$等距排列在一个圆上并将$G$的边画为直线，则$G$的图形具有$k$重旋转对称性，即图的所有信息被压缩到该图形的一个$360^\circ/k$的扇形楔形中。$G$中存在的最大$k$（使得$G$存在一个$k$-对称哈密顿圈）称为$G$的哈密顿压缩。我们研究了四类不同顶点传递图的哈密顿压缩，即超立方体图、约翰逊图、置换体图以及阿贝尔群的凯莱图。在若干情形中，我们精确确定了它们的哈密顿压缩；在其他情形中，我们给出了紧密的上下界。所构造的圈比文献中已知的若干经典格雷码具有更高的压缩率。我们的构造还生成了用于比特串、组合和排列的格雷码，这些格雷码具有少轨道和/或平衡性。