Log In Sign Up

The Hamilton compression of highly symmetric graphs

by   Petr Gregor, et al.

We say that a Hamilton cycle C=(x_1,…,x_n) in a graph G is k-symmetric, if the mapping x_i↦ x_i+n/k for all i=1,…,n, where indices are considered modulo n, is an automorphism of G. In other words, if we lay out the vertices x_1,…,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^∘/k wedge of the drawing. We refer to the maximum k for which there exists a k-symmetric Hamilton cycle in G 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 cycles we construct 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.


page 3

page 4

page 19

page 22

page 23

page 30

page 32


Flexible placements of graphs with rotational symmetry

We study the existence of an n-fold rotationally symmetric placement of ...

Codes with structured Hamming distance in graph families

We investigate the maximum size of graph families on a common vertex set...

Making an H-Free Graph k-Colorable

We study the following question: how few edges can we delete from any H-...

JPEG Noises beyond the First Compression Cycle

This paper focuses on the JPEG noises, which include the quantization no...

Graphs with the same truncated cycle matroid

The classical Whitney's 2-Isomorphism Theorem describes the families of ...

Compressing bipartite graphs with a dual reordering scheme

In order to manage massive graphs in practice, it is often necessary to ...