# Minimal Decompositions of Complete Graphs into Subgraphs with Embeddability Properties

@article{Beineke1969MinimalDO, title={Minimal Decompositions of Complete Graphs into Subgraphs with Embeddability Properties}, author={Lowell W. Beineke}, journal={Canadian Journal of Mathematics}, year={1969}, volume={21}, pages={992 - 1000} }

Although the problem of finding the minimum number of planar graphs into which the complete graph can be decomposed remains partially unsolved, the corresponding problem can be solved for certain other surfaces. For three, the torus, the double-torus, and the projective plane, a single proof will be given to provide the solutions. The same questions will also be answered for bicomplete graphs.

## 22 Citations

### On the bigenus of the complete graphs

- Mathematics
- 2021

We describe an infinite family of edge-decompositions of complete graphs into two graphs, each of which triangulate the same orientable surface. Previously, such decompositions had only been known…

### Current graphs and bi-embeddings

- MathematicsJ. Graph Theory
- 1978

The theory of current graphs is used to determine the values of N(γ,γ′) in certain cases, which is the size of the smallest complete graph which cannot be edge-partitioned into two parts embeddable in closed orientable surfaces of genera γ,β′ respectively.

### Planarizing Graphs - A Survey and Annotated Bibliography

- Computer Science, MathematicsJ. Graph Algorithms Appl.
- 2001

Given a nite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters.…

### The Thickness of Graphs: A Survey

- MathematicsGraphs Comb.
- 1998

A state-of-the-art survey of the thickness of a graph from both a theoretical and a practical point of view is given and some modifications of a basic heuristic are investigated for evaluating the thickness and determining a decomposition of agraph in planar subgraphs.

### The 4-girth-thickness of the complete graph

- MathematicsArs Math. Contemp.
- 2018

It is proved that the 4 -girth-thickness of an arbitrary complete graph K n, θ (4, K n ) , is ⌈( n +2)/4⌉ for n ≠‡‡6,‡10 and θ(4, K 6 )=3 .

### On the toroidal thickness of graphs

- MathematicsJ. Graph Theory
- 1982

T1(Gm), where Gm is the graph obtained from the complete graph Km by removing a Hamiltonian cycle, is found, and it is shown that t1(Kn(3)) = [1/2n] for many values of n.

### The 4-girth-thickness of the complete multipartite graph

- MathematicsElectron. J. Graph Theory Appl.
- 2019

The $4$-girth-thickness $\theta(4,G)$ of the complete $m$-partite graph $G$ when each part has an even number of vertices is calculated.

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A graph G consists of a finite set of p points and q lines joining pairs of these points. Each line joins two distinct points and no pair of points is joined by more than one line. A subgraph of G is…

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A graph consists of a finite set of points and a set of lines joining some pairs of these points. At most one line is permitted to join any two points and no point is joined to itself by a line. A…