^{1}

^{2}

Let
be a family of subgraphs of a graph G. An L-decomposition of G is an edge-disjoint decomposition of G into positive integer
copies of H
_{i}, where
. Let C
_{k}, P
_{k} and S
_{k} denote a cycle, a path and a star with k edges, respectively. For an integer
, we prove that a balanced complete bipartite multigraph
has a
-decomposition if and only if k is even,
and
.

Let F, G and H be graphs. A G-decomposition of F is a partition of the edge set of F into copies of G. If F has a G-decomposition, we say that F is G-decomposable. Let

For positive integers m and n,

Decompositions of graphs into k-stars have also attracted a fair share of interest (see [

Let G be a graph. The degree of a vertex x of G, denoted by

When

expresses the decomposition of G into

Lemma 1. ( [

Lemma 2. ( [

Let

Lemma 3. If k is an even integer with

Proof. A decomposition of

Note that

is,

Lemma 4. ( [

Lemma 5. ( [

Case | k | m | n | Conditions |
---|---|---|---|---|

1 | even | even | even | |

2 | even | even | odd | |

3 | even | odd | even | |

4 | odd | even | even | |

5 | odd | even | odd | |

6 | odd | odd | even | |

7 | odd | odd | odd |

With the results ( [

Main Theorem. Let k and n be positive integers. The graph

We first give necessary conditions for a

Lemma 6. If

Proof. Since bipartite graphs contain no odd cycle, k is even. In addition, the minimum length of a cycle and the maximum size of a star in

Throughout this paper, let

Lemma 7. For an even integer

Proof. Note that

Lemma 8. Let k be a positive even integer and let n be a positive integer with

Proof. Let

Case 1.

Let

Clearly

Let

and the subscripts of b are taken modulo k. Note that

interchange two edges

copies of

Let

The other case of t is odd, we have that

Let

with the center at

In the following, we will show that

of

is odd, that is, there exists an orientation of

where

where

We first consider the edges oriented outward from

From the construction of the orientation, it is easy to see that (2) and (3) are satisfied, and for all

So, we only need to check (1).

Since

for

Thus,

Therefore

Let

and each

Case 2.

Let

Then

Lemma 9. Let k be a positive even integer and let n be a positive integer with

Proof. Let

Trivially,

In the case of

Now we are ready for the main result. It is obtained by combining Lemmas 6, 7, 8 and 9.

Theorem 1. Let k and n be positive integers. The graph

Remark. Let m and n be positwe integers with

that

The authors are grateful to the referees for the helpful comments.

Jenq-Jong Lin,Min-Jen Jou, (2016) {C_{k}, P_{k}, S_{k}} -Decompositions of Balanced Complete Bipartite Multigraphs. Open Journal of Discrete Mathematics,06,174-179. doi: 10.4236/ojdm.2016.63015