A cycle partition or cycle cover of a graph is a

spanning subgraph such that each vertex is part of exactly one simple cycle.

A graph is said to be a subgraph of if and If ' contains all edges of that join two vertices in then is said to be the subgraph induced or spanned by , and is denoted by Thus, a subgraph of is an induced subgraph if If , then is said to be a

spanning subgraph of Two graphs are isomorphic if there is a correspondence between their vertex sets that preserves adjacency.

Basic routing, however, limits the traffic on the Ring Interval Graph, which is a

spanning subgraph of the original graph.

ns](H) where H is a

spanning subgraph of G, (iii) [[gamma].

A

spanning subgraph of G has the same set V of vertices.

Then H is a

spanning subgraph of some 2-tree on k - s - 2 vertices.

A graph H is a

spanning subgraph of G if V(H) = V(G) and E(H) [subset or equal to] E(G).

We apply this to prove that, for every d [greater than or equal to] 7 and every k, where 7 [less than or equal to] k [less than or equal to] d, the d-dimensional hypercube contains a k-regular

spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle.

We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d [greater than or equal to] 7 and every k, where 7 [less than or equal to] k [less than or equal to] d, the d-dimensional hypercube contains a k-regular

spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle.

a

spanning subgraph whose every connected component is a clique with at least 2 vertices.

Observation 1 Let G be a bridgeless graph and H be a bridgeless

spanning subgraph of G.

One consequence of the above theorem is that the restriction, in part 1 of the definition of a starter-matrix class C, that S be a labelled, induced subgraph of each element of C can be relaxed to S being a labelled subgraph of every graph in the collection: take the union of the starter-matrix classes such that S is a

spanning subgraph of the starter graph for the class.