Applied Mathematics
Vol.3 No.12(2012), Article ID:25347,3 pages DOI:10.4236/am.2012.312255
Some Results on (1,2n – 1)-Odd Factors
1Department of Basic Course, Air Force Logistics College, Xuzhou, China
2Aerial Four Station Department, Air Force Logistics College, Xuzhou, China
Email: liuman8866@163.com
Received September 4, 2012; revised November 1, 2012; accepted November 8, 2012
Keywords: Claw Free Graphs; -Odd Factor; Factor-Criticality
ABSTRACT
Let be a graph. If there exists a spanning subgraph
such that
, then
is called to be
-odd factor of
. Some sufficient and necessary conditions are given for
to have
-odd factor where
is any subset of
such that
.
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 thatTherefore
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, byConsequently
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.
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