Cube-connected cycles
Image:Cube-connected cycles.svg|thumb|upright=1.35|The cube-connected cycles of order 3, arranged geometrically on the vertices of a truncated cube.
In graph theory, the cube-connected cycles is an undirected cubic graph, formed by replacing each vertex of a hypercube graph by a cycle. It was introduced by for use as a network topology in parallel computing.
Definition
The cube-connected cycles of order n can be defined as a graph formed from a set of n2n nodes, indexed by pairs of numbers where 0 ≤ x < 2n and 0 ≤ y < n. Each such node is connected to three neighbors:,, and, where "⊕" denotes the bitwise exclusive or operation on binary numbers.This graph can also be interpreted as the result of replacing each vertex of an n-dimensional hypercube graph by an n-vertex cycle. The hypercube graph vertices are indexed by the numbers x, and the positions within each cycle by the numbers y.
Properties
The cube-connected cycles of order n is the Cayley graph of agroup that acts on binary words of length n by rotation and flipping bits of the word. The generators used to form this Cayley graph from the group are the group elements that act by rotating the word one position left, rotating it one position right, or flipping its first bit. Because it is a Cayley graph, it is vertex-transitive: there is a symmetry of the graph mapping any vertex to any other vertex.
The diameter of the cube-connected cycles of order n is for any n ≥ 4; the farthest point from is. showed that the crossing number of CCCn is + o) 4n.
According to the Lovász conjecture, the cube-connected cycle graph should always contain a Hamiltonian cycle, and this is now known to be true. More generally, although these graphs are not pancyclic, they contain cycles of all but a bounded number of possible even lengths, and when n is odd they also contain many of the possible odd lengths of cycles.