论文标题
高度对称图的汉密尔顿压缩
The Hamilton compression of highly symmetric graphs
论文作者
论文摘要
我们说，汉密尔顿周期$ c =（x_1，\ ldots，x_n）$ in Graph $ g $是$ k $ -smmetric，如果映射$ x_i \ mapsto x_ \ mapsto x_ {i+n/k} $用于所有$ i = 1，\ i = 1，\ ldots，n $，n $ n $ n $ n $ $ n $ g $ gom $ gom $ gom $ gomorphism y是自动$ gom的。换句话说，如果我们布置顶点$ x_1，\ ldots，x_n $在一个圆圈上均衡，并将$ g $的边缘绘制为直线，那么$ g $的绘图具有$ k $ fold的旋转对称性，即，有关该图的所有信息，所有有关图形的信息都被压缩为$ 360^\ 360^\ circ/k $ wedge of the Geedge of the Gudge of the Gredge of the Gudge of The Gudge of Geedge of Geedge of Geedge oh gedge of the图。存在$ g $的最大$ k $，$ g $中的$ k $ symmetric hamilton周期称为$ g $的汉密尔顿压缩。我们研究了四个不同家族的顶点传播图的汉密尔顿压缩，即超级立管，约翰逊图，Permutahedra和Abelian组的Cayley图。在某些情况下，我们会准确确定他们的汉密尔顿压缩，在其他情况下，我们提供了近距和上限。构造的循环的压缩比文献中已知的几种经典的灰色代码要高得多。我们的构造还产生了灰色代码，用于几乎没有轨道和/或平衡的轨迹，组合和排列。
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.
