Search

A graph is k -cycled if all its cycles are integral multiples of an integer k ≥ 2. We determine the structure of refinements of Kn and Kn, m which are k-cycled.

Homeomorphic embeddings of Kn in the m-cube were investigated in [6]. In particular, it was proved that any homeomorph of Kn+1 embedded in the m-cube has at least n2 edges. Furthermore, homeomorphic embeddings of Kn+1 having exactly n2 edges are unique up to isomorphism. In this paper a similar problem for the complete bipartite graph is considered.

We adopt the notation and terminology of [5].

All graphs considered are without loops and multiple edges.

Let x = uv be an edge of a graph G; x will be called subdivided if it is replaced by a vertex ω and by edges uω and ωv. A graph G’ is called a subdivision of G if it is obtained from G by a subdivision of an edge of G. A refinement Ĝ of G, is a graph isomorphic to a graph obtained from G by a finite sequence of subdivisions.

The coarseness, c(G), of a graph G is the maximum number of edge disjoint nonplanar subgraphs contained in G For the n-dimensional cube Q n we obtain the inequalities

Search Results

Refine search

Refine search

Actions for selected content:

3 results

On k-Cycled Refinements of Certain Graphs

On Homeomorphic Embeddings of Km,n in the Cube

Bounds on the Coarseness of the n-Cube

Search Results

Refine search

Refine search

Actions for selected content:

Save Search

3 results

On k-Cycled Refinements of Certain Graphs

On Homeomorphic Embeddings of Km,n in the Cube

Bounds on the Coarseness of the n-Cube