  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.
