1. Introduction
Let G be a graph with vertex set
and edge set
. An external bipartition of G is
and
and requires that at least half of the neighbors of each vertex are in the other part. If a bipartition satisfies
, we call it a bisection and denote it by
. Ban and Linial [1] showed that every class 1, 3 or 4 regular graph G has an external bisection. Bollobás and Scott [2] observed that not every graph has an external bisection. In the same paper, they gave a counterexample that
, where
doesn’t have an external bisection. Esperet, Mazzuoccolo and Tarsi [3] found a set of cubic graphs without external bisection and containing at least 2 bridges.
For vertices
, if
, then edge uv is said to be associated with u and v. The degree of v is the number of edges in G associated with v, denoted as
. Let
be a bisection of G. For a vertex v in V1, the internal degree of v is the number of the edges associated with v and other endpoints of these edges are in V1 too; and the external degree of v is the number of the edges associated with v and other endpoints of these edges are in V2. For a vertex v in V2, the internal degree of v is the number of the edges associated with v and other endpoints of these edges are in V2; and the external degree of v is the number of the edges associated with v and other endpoints of these edges are in V1. The internal degree of v is denoted as
and the external degree of v is denoted as
.
A graph is said to be a bipartite graph, denoted by
, if the set of vertices of the graph can be partitioned into two non-empty subsets X and Y such that no two vertices in X are connected to each other with an edge and no two vertices in Y are connected with an edge.
The crown graph of [4]
satisfies the condition:
;
For example, Figure 1 shows
.
A windmill graph
is a graph consisting of n m-order complete graphs
with a common vertex.
For example, Figure 2 shows
.
Conjecture 1.1. (Bollobás and Scott [2] ). Every graph G has a weak external bisection that is G has a bisection
, such that:
for all
.
Ji, Ma, Yan and Yu [5] showed that every graphic sequence has a realization for which Conjecture 1.1 holds. In the same paper, they gave an infinite family of counterexamples to Conjecture 1.1.
In this paper, we confirm this conjecture for some graphs by showing the following three theorems.
Theorem 1.1. Let
be a bipartite graph with
, then
admits a weak external bisection.
Theorem 1.2. Every crown graph
admits a weak external bisection.
Theorem 1.3. Every windmill graph
admits a weak external bisection.
In fact, by the proof Theorem 1.2,
admits an external bisection.
2. Weak External Bisection
In this section, we prove Theorem 1.1, Theorem 1.2 and Theorem 1.3.
Proof of Theorem 1.1. Let
be a bipartite graph with two parts X and Y, we define
.
For a set S, we define a function:
(1)
Let
, and assume, without loss of generality, that:
.
Otherwise, we can re-label the three vertices in X. We give a weak external bisection
of
by three steps.
First, let
,
,
and
where
and
. Because
, such
exists. Let
.
Then, we partition the odd sets of
,
,
,
,
,
one after another. Denote by
, the odd sets of
,
,
,
,
,
, with the same order. For each
, put
vertices of
into V2 and
vertices of
into V1 if i is odd; and put
vertices of
into V1 and
vertices of
into V2 if i is even.
Finally, denote by
the even sets of
,
,
,
,
,
. For each
, put
vertices of
into V1 and
vertices of
into V2.
Now, we show that V1 and V2 form a weak external bisection of
. Clearly,
if m is odd, and
if m is even. So
is a bisection. Since
and
, then
for each vertex
. Moreover,
for each vertex
if
and
or 3; and
for each vertex
if
,
and
. So, for each
,
.
Let:
Let
, for each
. Note that
are the odd sets of
,
,
,
,
,
, with the same order. Then, each
contains several continuous set of
. Let
, for each
. We have:
Since
contains continuous set of
, then we see that
, where
is defined as (1). It is easy to see that:
Clearly, if
is odd, then
. Thus, we have
. Similarly, we have:
So, we have
. Then, for each
,
. Thus,
is a weak external bisection of
.n
Proof of Theorem 1.2. Let
. We give a weak external bisection
of
by two steps.
First, let
and
. Let
and
.
Then, we partition the set
. The partition of
is determined by
. For a given k, if
, let
and
; if
, let
and
.
Now, we show that V1 and V2 form a weak external bisection of
. Clearly,
if both n and m are odd. Otherwise,
. So,
is a bisection. If n is even, then
for each
. If n is odd, then
,
and
for each
. So, in any case,
for
.
If m is odd, then
for each
,
for each
and
for each
. If m is even, then
,
,
for each
and
for each
. So, in any case,
for
;
.
If both n and m are odd, then
,
and
for each
. If n is odd and m is even, then
,
and
for each
. If n is even and m is odd, then
for each
. If both n and m are even, then
for each
. So, in any case,
for
. Thus,
is a weak external bisection of
.n
Proof of Theorem 1.3. We labeled the vertices of graph
as in Figure 2.
. Let
.
We consider the following two cases.
Case 1. m is odd.
Let
and
. Now
. So
is a bisection.
. For each
,
. For each
,
. Thus,
is a weak external bisection of
.
Case 2. m is even.
If i is odd, let
and let
. If i is even, let
and let
.
Now, we show that V1 and V2 form a weak external bisection of
. Clearly,
if n is odd and
if n is even. So
is a bisection. If i is odd,
for each
; and
for each
. If i is even, then
for each
; and
for each
. If n is even, then
. If n is odd, then
. Thus,
is a weak external bisection of
.n