1. Introduction
We consider finite undirected graph without loops and multiple edges .Let 
be a graph with vertex set 
and edge set 
Given
, the set of vertexes adjacent to 
is said to be the neighborhood of
, denoted by
, and 
is called the degree of
, If there exists a spanning subgraph 
such that
, then 
is called a 
-odd factor of
, especially, if for every 
such that
, then it is called 
-odd factor. especially, 
-odd factor is 1-factor when n = 1. For a subset
, let 
denote the subgraph obtained from 
by deleting all the vertexes of 
together with the edges incident with the vertexes of 
denotes the number of odd components of
. The sufficient and necessary condition for graph to have 
-odd factor was given in paper [1] Ryjacek [2] introduced one kind of new closure operation: let 
be a graph, 
, if the subgraph induced by 
is not complete graph, we consider the following operation: jointing every pair of nonadjacent vertex in 
makes 
to be a complete graph. The operation is called local completely at point
. If the subgraph induced by 
is k-vertex connected, then vertex 
is called local 
-vertex connected graph
.
Favaron gave the concept of k-factor critical in paper [3]. If
, and for any 
, 
is perfect matching ,then we call the graph 
to be k-factor critical. Of course, 0-factor critical graph is perfect matching. Favaron popularized a series of the properties of perfect matching to k-factor critical, at the same time the sufficient and necessary conditions were given for the graph to be k-factor critical, more results in factor critical graphs were referred to [4,5].
For 
-odd factor, Chen Ci-ping [6] gave a sufficient condition for a matching with exactly 
edges extended to 
-odd factor. Teng Cong generalized some results on kto 
-odd factor, and proved that the connected graph 
exists 
-odd factor with k-extended, then for the any edge 
of
, 
exists
-odd factor [7] with 
-extended. If there exists 
-odd factor of 
with k-extended, then there exists 
-odd factor with 
-extended, and 
is 
-connected [8]. We will popularize some results of k-factor critical to 
-odd factor, and gain several sufficient and necessary conditions for 
to have 
-odd factor for any subset 
of 
such that
.
2. Main Results
We start with some lemmas as following.
Lemma 1 The sufficient and necessary condition for a graph 
to have 
-odd factor after cutting off any 
vertexes is
Proof For set 
with any 
vertexes, 
has 
-odd factor, next we will prove
For any 
and
, let
, where
. Since 
has a 
-odd factor, by the sufficient and necessary condition for graph with 
-odd factor we have
.
Noting that
Therefore
For any 
and 
we have
the following that the set 
with any 
vertexes, 
has 
-odd factor, i.e., for any
, there
.
Noting that
, of course
.
By
and
, we have
Lemma 2 [9] Connected claw free graphs of even order have 1-factor.
Lemma 3 Connected claw free graphs of even order have 
-odd factor.
Proof If
, by lemma 2, the conclusion is proved. Assume that
.
By contradiction, we assume that 
has no 
-odd factor, i.e., 
such that
.
then there exists 
such that 
connecting with three components of 
at least. If not, for
, 
connects with two components of 
at most, consequently
, contradiction.
Theorem 1 Let 
be graph with 
order, 
are a couple of nonadjacent vertexes and satisfy
then the sufficient and necessary condition for 
removing any 
vertexes with 
-odd factor is that 
getting rid of any 
vertexes with 
-odd factor.
Proof The necessary condition is obvious, next we prove the sufficient condition.
By contradiction, let 
remove any 
vertexes with 
-odd factor, but there exist 
vertexes after getting rid of the 
vertexes of 
without 
-odd factor. By lemma 1, there exists
such that
and
.
at the same time, by 
and
Thereby
.
Furthermore, by
Consequently
Accordingly
and
.
It shows that 
are part of two odd components 
of 
respectively.
Thus
.
On the other hand, by hypothesis
But
.
Contradiction.
Theorem 2 Let 
connected graph 
be 
order, 
are a couple of any nonadjacent vertexes of
, and satisfy
then the sufficient and necessary condition for 
removing any 
vertexes with 
-odd factor is 
getting rid of any 
vertexes with
-odd factor.
Proof 
is a spanning subgraph of
, so the necessary condition is obvious.
Next we prove the sufficient condition. We suppose 
getting rid of any 
vertexes with 
- odd factor, but 
is not, i.e. there exist
such that
.
Be similar to the discussion of theorem 1
and
.
thereby 
are part of two odd components 
of 
respectively.
Noting that
(1)
By hypothesis
(2)
Combining (1) with (2)
Consequently
but
.
Contradiction.
Theorem 3 Let 
be claw free graphs, 
be partial 
connection point. 
be graph obtained by locally fully on 
in 
point, then for
, the sufficient and necessary condition for 
with 
-odd factor is 
with 
-odd factor.
Proof 
is a spanning subgraph of
, so the necessary condition is obvious.
Next we prove the sufficient condition. Let 
have 
-odd factor, 
have no 
- odd factor. 
has 
-odd factor, 
, so
.
On the other hand, 
is claw free, so 
is claw free.
By lemma 2, lemma 3, 
has two odd components at least.
If
, let 
(
is branch of
). Now, 
has the same odd components as
, therefore, 
has 
-odd factor. which is contradiction.
Next let
, since 
has not odd components, for any odd components of
,
is complete.
Let 
be adjacent vertexes of 
in two odd components of 
respectively.
Then 
is nonadjacent in the induced subgraph of
, which is contradiction to the fact that
is a locally 
connected vertex, since
The proof is complete.