The Hamilton compression of highly symmetric graphs

TL;DR

This paper introduces the concept of Hamilton compression, measuring the symmetry-based data compression of Hamilton cycles in highly symmetric graphs, and explores this property in various well-known graph families.

Contribution

It defines Hamilton compression for graphs and determines its values for hypercubes, Johnson graphs, permutahedra, and Cayley graphs, providing exact results and bounds.

Findings

Hamilton cycles with high symmetry are constructed in various graph families.

Hamilton compression often exceeds classical Gray code efficiencies.

New Gray codes with fewer tracks and balanced properties are derived.

Abstract

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.