Open Journal of Discrete Mathematics
Vol.06 No.03(2016), Article ID:68338,6 pages
10.4236/ojdm.2016.63015
-Decompositions of Balanced Complete Bipartite Multigraphs
Jenq-Jong Lin1, Min-Jen Jou2
1Department of Finance, Ling Tung University, Taiwan
2Department of Information Technology, Ling Tung University, Taiwan
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
Received 16 June 2016; accepted 11 July 2016; published 14 July 2016
ABSTRACT
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
, where
. Let
,
and
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
.
Keywords:
Balanced Complete Bipartite Multigraph, Cycle, Path, Star, Decomposition
1. Introduction
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 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
, where
. If G has an L-decomposition, we say that G is L-decomposable.
For positive integers m and n, denotes the complete bipartite graph with parts of sizes m and n. A complete bipartite graph is balanced if
. A k-cycle, denoted by
, is a cycle of length k. A k-star, denoted by
, is the complete bipartite graph
. A k-path, denoted by
, is a path with k edges. For a graph G and an integer
, we use
to denote the multigraph obtained from G by replacing each edge e by
edges each of which has the same ends as e.
Decompositions of graphs into k-stars have also attracted a fair share of interest (see [1] - [3] ). Articles of - decompositions of interest include [4] [5] . Decompositions of some families of graphs into k-cycles have been a popular topic of research in graph theory (see [6] [7] for surveys of this topic). The study of
-decom- position was introduced by Abueida and Daven in [8] . Abueida and Daven [9] investigated the problem of
-decomposition of the complete graph
. Abueida and O’Neil [10] settled the existence problem for
-decomposition of the complete multigraph
for
. In [11] , Priyadharsini and Muth- usamy gave necessary and sufficient conditions for the existence of a
-factorization of
where
. Furthermore, Shyu [12] investigated the problem of decomposing
into paths and stars with k edges, giving a necessary and sufficient condition for
. In [13] , Shyu considered the existence of a decomposition of
into paths and cycles with k edges, giving a necessary and sufficient condition for
. Shyu [14] investigated the problem of decomposing
into cycles and stars with k edges, settling the case
. Recently, Lee [15] [16] established necessary and sufficient conditions for the existence of a
-decomposition of a complete bipartite graph and
-decomposition of a balanced complete bi- partite graph. Lin and Jou [17] investigated the problems of the
-decomposition of the balanced complete bipartite graph
. It is natural to consider the problem of the
-decomposition of the balanced complete bipartite multigraph
for
. In this paper, the necessary and sufficient conditions for the existence of such decomposition are given.
2. Preliminaries
Let G be a graph. The degree of a vertex x of G, denoted by, is the number of edges incident with x. The vertex of degree k in
is the center of
. For
and
, we use
and
to denote the subgraph of G induced by A and the subgraph of G obtained by deleting B, respectively.
When are graphs, not necessarily disjoint, we write
or
for the graph with vertex set
and edge set
. When the edge sets are disjoint,
expresses the decomposition of G into.
is the short notation for the union of n copies of disjoint graphs isomorphic to G. Let H be a subgraph of
with vertex set
and edge set
, and let r be a nonnegative integer. We use
to denote the graph with vertex set
and edge set
where the subscripts of b are taken modulo n. For any vertex x of a digraph G, the outdegree
(respectively, indegree
) of x is the number of arcs incident from (respectively, to) x. A multistar is a star with multiple edges allowed. We use
to denote a multistar with k edges. Let G be a multigraph. The edge-multiplicity of an edge in G is the number of edges joining the vertices of the edge. The multiplicity of G, denoted by
, is the maximum edge- multiplicity of G.
Lemma 1. ( [3] ) For integers m and n with, the graph
has an
-decomposition if and only if
and
Lemma 2. ( [18] ) Suppose that. Then
is
-decomposable.
Let denote the edge ab in the s-th copy
of
for
.
Lemma 3. If k is an even integer with, then there exist
edge-disjoint 2k-cycles in
.
Proof. A decomposition of into 2k-cycles is given by the following
cycles:
, where
,
and
. □
Note that can be decomposed into two copies of k-paths:
and
, that
is, can be decomposed into
copies of k-paths.
Lemma 4. ( [4] ) There exists a -decomposition of
if and only if
, and one of the following (see Table 1) cases occurs.
Lemma 5. ( [19] ) For positive integers m, n and k, the graph has a
-decomposition if and only if
and k are even,
,
, and
.
Table 1. The conditions of a -decomposition of
.
3. Main Results
With the results ( [17] ) of the -decomposition of the balanced complete bipartite graph
, it is assumed that
in the sequel. In this section, we will prove the following result.
Main Theorem. Let k and n be positive integers. The graph has a
-decomposition if and only if k is even,
and
.
We first give necessary conditions for a -decomposition of
.
Lemma 6. If has a
-decomposition, then k is even,
and
.
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 are 4 and n, respectively, we have
. Finally, the size of each member in the decomposition is k and
; thus
. □
Throughout this paper, let denote the bipartition of
, where
and
. We now show that the necessary conditions are also sufficient. The proof is divided into cases
,
, and
, which are treated in Lemmas 7, 8, and 9, respectively.
Lemma 7. For an even integer, then
has a
-decomposition.
Proof. Note that. By Lemmas 1 and 4,
has a
-decomposition and a
-decomposition. In addition, by Lemma 5,
has a
-decomposition. Hence
has a
-decomposition. □
Lemma 8. Let k be a positive even integer and let n be a positive integer with. If
is divisible by k, then
has a
-decomposition.
Proof. Let. From the assumption
, we have
. Let
. Since
, we have
, which implies that t is a positive integer. The proof is divided into two cases according to the values of t.
Case 1..
Let,
,
and
.
Clearly. Note that G is isomorphic to
,
is isomorphic to
,
is isomorphic to
and F is isomorphic to
, which can be decomposed into
copies of
by Lemmas 1 and 2. In the following, we will show that
can be decomposed into
copies of
, one copy of
and
copies of
.
Let, where
and
. Define a subgraph W of G as follows:
and the subscripts of b are taken modulo k. Note that for t is even, and
for t is odd, this assures us that there are enough edges for W. Note that a
can be decomposed into 2 copies of
. In addition,
for t is even as well as
for t is odd, it follows that W can be decomposed into t copies of
. Since
, we
interchange two edges in
and
in
, then we obtain a new cycle
. Hence
can be decomposed into
copies of and one copy of
.
Let be the graph obtained from G by deleting the edges in W. For the case of t is even, we have that
The other case of t is odd, we have that
Let for
. Then for t is even
, and for t is odd
with the center at.
In the following, we will show that can be decomposed into r copies of
with centers in
, and into k copies of
with centers in
for t is even as well as k/2 copies
of with centers in
and
copies of
with centers in
for t
is odd, that is, there exists an orientation of such that
(1)
where, and for t is even
(2)
where, and for t is odd
(3)
We first consider the edges oriented outward from. If t is even, then the edges
are all oriented outward from
where
. If t is odd, for
, the edges
and
are all oriented outward from
, where the subscripts of b are taken modulo r in the set
. In both of the cases the subscripts of b are taken modulo r in the set of numbers
. Since
for t is even, and
for t is odd, this assures us that there are enough edges for the above orientation. Finally, the edges which are not oriented yet are all oriented from
to
.
From the construction of the orientation, it is easy to see that (2) and (3) are satisfied, and for all, we have
(4)
So, we only need to check (1).
Since for
, it follows from (4) that
for. Note that t is even,
, and t is odd,
Thus,
Therefore for
. This proves (1). Hence, there exists the required decomposition
of
. Let
be the star with center at
in
for
. Then
is an
. Since
, by Lemma 2, we obtain that
can be decomposed into
copies of
for
.
Let be the
-multistar with center at
in
for
. Let
for
. Then
is decomposed into
,
,
and each. It follows that
. Since
, by Lemma 2, we obtain that
can be decomposed into
copies of
for
. Recall that
, we have that
is
-decomposable.
Case 2..
Let,
,
and
.
Then. By similar arguments as in the proof of Case 1, we have that
can be decomposed into one copy of
and
copies of
. On the other hand, by Lemma 5,
has a
-decomposition. Hence
has a
-decomposition. □
Lemma 9. Let k be a positive even integer and let n be a positive integer with. If
is divisible by k, then
has a
-decomposition.
Proof. Let where q and r are integers with
. From the assumption of
, we have
. Note that
Trivially, ,
and
are multiples of k. Thus
from the assumption that
is divisible by k. By Lemmas 1 and 2,
,
and
have
-decomposition.
In the case of, by Lemma 7, we obtain that
has a
-decomposition. In addition, by Lemma 8,
has a
-decomposition for
. Hence there exists a
-de composition 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 has a
-decomposition if and only if k is even,
and
.
Remark. Let m and n be positwe integers with. 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
are 4 and m, respectively, we have
. Moreover, each k-cycle in
uses
vertices of each partite set, which implies
that. Finally, the size of each member in the decomposition is k and
, thus
. Hence the obvious necessary conditions for the graph
to have a
-de composition are: 1) k is even, 2)
, and 3)
. It is natural to ask whether they are sufficient.
Acknowledgements
The authors are grateful to the referees for the helpful comments.
Cite this paper
Jenq-Jong Lin,Min-Jen Jou, (2016) {Ck, Pk, Sk} -Decompositions of Balanced Complete Bipartite Multigraphs. Open Journal of Discrete Mathematics,06,174-179. doi: 10.4236/ojdm.2016.63015
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