doi:10.4236/iim.2010.22017 Published Online February 2010 (http://www.scirp.org/journal/iim)

**Signed (b,k)-Edge Covers in Graphs **

**A. N. Ghameshlou1 _{, Abdollah Khodkar}2_{, R. Saei}3_{, S. M. Sheikholeslami}**

_{*}

**3**

1* _{Department of Mathematics, University of Mazandaran Babolsar, I.R., Iran }*
2

*3*

_{Department of Mathematics, University of West Georgia, Carrollton, USA }

_{Department of Mathematics, Azarbaijan University of Tarbiat MoallemTabriz, I.R., Iran }*Email*:*akhodkar@westga.edu, s.m.sheikholeslami@azaruniv.edu *

**Abstract **

Let be a simple graph with vertex set and edge set . Let have at least vertices of

degree at least , where and *b* are positive integers. A function is said to be a

signed -edge cover of *G* if

*G* *V G*( )

( )

*e E v*

( )

*E G* *G*

: (
*f E*

*k*

*b* *k* *G*) { 1,1}

( , )*b k*

###

_{}

*f e*( )

*b*for at least vertices of , where . The value

*k* *v* *G*

( ) = {*uv E*( ( )

*E v* *G u N v*) | } min

###

_{e E}_{}

_{( )}

_{G}*f e*( ), taking over all signed ( , )

*b k*-edge covers

*f*of

*G*

is called the signed ( , )*b k* -edge cover number of *G* and denoted by _{b k}_{,} ( )*G* . In this paper we give some
bounds on the signed ( ,*b k*)-edge cover number of graphs.

**Keywords: **Signed Star Dominating Function, Signed Star Domination Number, Signed -edge Cover,

Signed -edge Cover Number

( , )*b k*
( , )*b k*

**1. Introduction **

Structural and algorithmic aspects of covering vertices
by edges have been extensively studied in graph theory.
An *edge cover* of a graph G is a set C of edges of G such
that each vertex of G is incident to at least one edge of C.
Let b be a fixed positive integer. A *b-edge cover* of a
graph G is a set C of edges of G such that each vertex of
G is incident to at least b edges of C. Note that a *b*-edge
cover of G corresponds to a spanning subgraph of G with
minimum degree at least b. Edge covers of bipartite
graphs were studied by König [1] and Rado [2], and of
general graphs by Gallai [3] and Norman and Rabin [4],
and *b*-edge covers were studied by Gallai [3]. For an
excellent survey of results on edge covers and *b*-edge
covers, see Schrijver [5].

We consider a variant of the standard edge cover
problem. Let *G* be a graph with vertex set and
edge set . We use [6] for terminology and notation
which are not defined here and consider only simple
graphs without isolated vertices. For every nonempty
subset of , the subgraph of *G* whose vertex
set is the set of vertices of the edges in

( )
*V G*
( )

*E G*

*E* *E G*( )

*E*

###

and whoseedge set is *E*, is called the subgraph of *G* induced by
*E* and denoted by [ ]*G E* . Two edges of
are called *adjacent* if they are distinct and have a
common vertex. The *open neighborhood* of an
edge

1, 2

*e e*

( )
*G*
*N e*

*G*

( )*G*

*e E* is the set of all edges adjacent to . Its
*closed neighborhood* is . For a
function and a subset of we
define

*e*

) [ ] =

*N e _{G}*
R

( )

( ){ }*e*
(
*E G*
*G*

*N e*
*S*
: (

*f E*
( )

)
*G*

*e S*
=

*f S*

###

_{}

*f e*. The

*edge-neighborhood*of a vertex

( )*v*
*G*

*E* *v V* (*G*) is the set of all edges
incident to vertex

*v*

. For each vertex , we
also define
(
*v V* *G*)

( )

*e E _{G}*

*v*

( ) = ( )

*f v*

###

_{}

*f*

*e*. Let be a positive

integer and let have at least vertices of degree at
least . A function is called a
*signed * *-edge cover* (SbkEC) of *G*, if

*b*

{ 1

*G* *k*

)

*G*

*b*
( ,*b k*

: (

*f E* ,1}

) *f v*( )*b*

for at least vertices of . The *signed *
*-edge cover number* of a graph G is

*k*
)

( ) =*G*

*v*

*G*

( ,*b k*

,

*b k* min

{

###

_{e E}_{}

*f e*( ) |

*f is an SbkEC on G*}. The signed ( , )

*b k*-edge cover

*f*of

*G*with

,

= *b k*
( ( )) ( )

*f E G* *G* is called a *b k*, ( )*G* -*cover*. For any

signed ( , )*b k* -edge cover *f* of *G* we define

= { | ( ) = 1}
*P* *e E f e*

= { | ( ) }
*V* *v V f v*_{}

= 1

*b* *k*=

, ,

and .

= { | ( ) = 1}
*M* *e E f e*

= { | ( ) < }
*V* *v V f v*_{} *b*

( , )*b k*

'( )
*SS* *G*
*b*

If and , then the signed -edge
cover number is called the * signed star domination *
*number*. The signed star domination number was intro-
duced by Xu in [7] and denoted by

*n*

. The signed star domination number has been studied by several authors (see for example [7,10]).

If and 1 , then the signed -edge
cover number is called the *signed star * *-subdomination *
*number*. The signed star -subdomination number was
introduced by Saei and Sheikholeslami in [11] and
denoted by .

= 1
*b*

( )
*k*
*SS* *G*
*b*

( , )*b k*
*b*

( )
*b* *G*

*k*
*n*

*k*

( , )*b k*
*k*

=
*k* *n*
If is an arbitrary positive integer and , then
the signed -edge cover number is called the
*signed -edge cover number*. The signed *b*-edge cover
number was introduced by Bonato *et al.* in [12] and
denoted by

( , )*b k*
.

The purpose of this paper is to initiate the study of the
signed -edge cover number _{b k}_{,} ( )*G*

'( )
*SS* *G*

. Here are
some well-known results on , *k* ( ) and

*SS* *G*
( )

*b* *G*

1,*n*'( ) 2*G* *n* 4
.

**Theorem 1** [10] For every graph G of order *n*4,

.

**Theorem 2** [11] For every graph of order
without isolated vertices,

*G* *n*

'( ) 4.

*k* *G* *n k*

4

1,

**Theorem 3** [10] For every graph *G* of order

without isolated vertices,

*n*

1,

'( )
2
*n*

*n*
*G*

*G* *n*

.

**Theorem 4** [11] For every graph of order
without isolated vertices,

2

1,*k*'( )*G*

*G*
*b*

( ( ) 1) ( ) 2

*G* *k n G*

_{}

*n*

.

**Theorem 5** [12] Let *b* be a positive integer. For
every graph of order and minimum degree at
least ,

,

*b n*'( ) .
2
*bn*
*G*

We make use of the following result in this paper.
**Theorem 6** [7] Every graph *G* with ( ) 3 *G*
contains an even cycle.

**2. Lower Bounds for SbkECN of Graphs **

In this section we present some lower bounds on

terms of the order, the size, the maximum degree and the degree sequence of . Our first proposition is a gene- ralization of Theorems 3, 4 and 5.

*G*

**Proposition 1** Let be a graph of order without
isolated vertices and maximum degree . Let
be a positive integer and let be the number of
vertices with degree at least . Then for every positive
integer

*G* *n*

= (
*G*)

*b* *n*_{0} 1

*b*

0

1 *k n* ,

0 ,

( ) ( 1) ( 1

( ) .

2
*b k*

*k b* *n* *b* *n b*

*G*

)

*Proof.* Let *f* be a *b k*, ( )*G* -cover. We have

,

( ) ( ) ( )

( ) ( )

0 0

0

1

( ) = ( ) = ( )

2

1 1

= ( )

2 2

( ) ( )( 1)

2 2

( ) ( 1) ( 1)

= .

2
*b k*

*e E G* *v V G e E v*

*e E v* *e E v*

*v V* *v V*

*G* *f e* *f e*

( )

*f e* *f*

*n* *k* *n n b*

*kb*

*k b* *n* *b* *n b*

###

###

###

###

*e*

**Theorem 2** Let be a graph of order , size ,
without isolated vertices and with degree sequence

, where

*G* *n* *m*

1 2

( , , , )*d d* *d _{n}*

*d*

_{1}

*d*

_{2}

*d*

_{n}0 1

*n*

. Let *b* be a
positive integer and let be the number of vertices
with degree at least . Then for every positive integer *b*

0

1 *k n* ,

2 =1

,

( )

( ) .

2
*k*

*j* *j*
*j*

*b k*

*n*
*bd* *d*

*G* *m*

*d*

###

*Proof.* Let *g* be a _{b k}_{,} ( )*G* -cover of *G* and let
( )

*g v* *b*
{ , ,*v*

for distinct vertices *k* *v* in *B G*( ) =
( ) =

1 }

*j* *v _{jk}* . Define

*f E G*: ( ){0,1} by

*f e*( ( ) 1) / 2

*g e* for each

*e E G* ( ). We have

( ) = ( )

( [ ]) deg( ) deg( ) 1 ( [ ]) =

2
*G*

*G*

*e E G* *e uv E G*

*g N e* *u* *v*

*f N e*

###

###

.(1)

,

*b k*

( )

( ( *G*[ ]) ( )) = ( ( )) deg( )

*e E G* *v V*

*g N e* *g e* *g E v* *v*

###

###

and

2 = ( )

(deg( ) deg( )) = deg( ) ,

*e uv E G* *v V*

*u* *v* *v*

###

###

by (1) it follows that

( )

( )

\{ , , } 1

2 , =1

2 , =1

2 , =1

( [ ])

1 1

deg( )( ( ( )) deg( )) ( )

2 2 2

1

deg( )( ( ( )) deg( )) 2

1 _{(} _{)} 1 _{( )}

2 2 2

1 _{(} _{)} 1 _{( )}

2 2 2

1 _{(} _{)} 1 _{( )} _{.}

2 2 2

*G*
*e E G*

*v V* *e E G*

*v V v _{j}*

*v*

*jk*

*k*

*j _{i}*

*j*

_{i}*b k*

*i*

*k*

*j _{i}*

*j*

_{i}*b k*

*i*

*k*

*j* *j* *b k*
*j*

*f N e*

*m*

*v g E v* *v* *g e*

*v g E v* *v*

*m*

*bd* *d* *G*

*m*

*bd* *d* *G*

*m*

*bd* *d* *G*

###

###

###

###

###

###

###

(2)

On the other hand,

( ) ( )

( )

( ) ( )

( )

( [ ]) = ( ( )) deg( ) ( )

( ( )) ( )

= (2 ( )) ( )

= (2 1) ( ).
*G*

*e E G* *v V* *e E G*

*n*

*v V* *e E G*

*n*

*e E G* *e E G*

*n*

*e E G*

*f N e* *f E v* *v* *f e*

*f E v d* *f e*

*d* *f e* *f e*

*d* *f e*

###

###

###

###

###

###

###

###

(3)

By (2) and (3)

2 , =1

( )

1 _{(} _{)} 1 _{( )}

2 2

( ) .

2 1

*k*

*j* *j* *b k*
*j*

*e E G* *n*

*m*

*bd* *d* *G*

*f e*

*d*

###

###

2 (4)Since ( ( )) = 2 ( ( ))*g E G* *f E G* *m*, by (4)

, ( ) = ( )

*b k* *G* *g e*

###

2 , =1

1

( ( ) ( ) ) .

2 1

*k*

*j* *j* *b k*
*j*

*n*

*bd* *d* *G* *m* *m*

*d*

###

Thus,

2 =1

,

( )

( ) ,

2
*k*

*j* *j*
*j*

*b k*

*n*
*bd* *d*

*G* *m*

*d*

###

as desired.

An immediate consequence of Theorem 2 is:

**Corollary 3 **For every

*r*

-regular graph *G*of size

*m*,

,

( )

( ) .

2
*b k*

*k b r*

*G* *m*

Furthermore, the bound is sharp

for -regular graphs with *r* *b r*= and *k*=*n*.

**Theorem 4** Let *G* be a graph of order , size ,
without isolated vertices, with minimum degree

2

*n* *m*

= ( )*G*

0

1 *k n*

and maximum degree . Let *b* be a
positive integer and be the number of vertices
with degree at least *b*. Then for each positive integer

= (
*G*)

0 1

*n*

, '( )

*b k* *G*

2 2 2 2 2

0

( ) 2( ) ( 1) ( ( 1) )

. 2

*b k* *m* *b* *n* *b* *n*

(5)

Furthermore, the bound is sharp for -cycles when and .

*n*

= 2
*b* *k*=*n*

*Proof.* Let *B G*( ) = {*v V G* ( ) | deg( )*v* *b*}
)

and let be a

*f*

, (

*b k* *G*

-cover. Since for each , ,

it follows that

*v V*_{} _{f v}_{( )}_{}_{b}

deg( ) | ( ) |

2

*v* *b*

*M* *E v* . Thus

=

=

( [ ])

2

2

(2 1) | |

(deg( ) deg( ) 1)

| | (deg( ) deg( ))

| | | ( ) | deg( )

| | | ( ) | deg( ) | ( ) | deg( )

deg( )

| | deg( ) deg( )

2

deg( ) | |

2

###

###

###

###

###

###

###

###

*e uv M*

*e uv M*

*v V G M*

*v V* *v V*

*v V* *v V*

*M*

*u* *v*

*M* *u* *v*

*M* *M* *E v* *v*

*M* *M* *E v* *v* *M* *E v* *v*

*v* *b*

*M* *v* *v*

*v*

*M* _{deg( )}2 _{deg( )}

2

_{}

2 2 2

2 ( ) 2 2

\ ( )

2 2 2

2 2 2

0 0

deg( ) deg( )

| | | |

2 2 2

deg( ) deg( ) | |

2 2

deg( ) _{|} _{|}

2 2

( 1)

| | | ( ) | | \ ( )

2 2 2

( 1)

| | ( ) ( )

2 2 2

*v V* _{v V}

*v V* _{v V}_{B G}

*v V* *B G*

*v* *v* *b*

*M* *V*

*v* *v*

*M*

*v* *b*
*V*

*b* *b* _{|}

*M* *m* *k* *V* *B G* *V* *B G*

*b* *b*

*M* *m* *k* *n* *k* *n n*

_{}

_{} _{}

###

###

###

###

###

Hence,

2 2 2 2 2 0

1

| | (( 1) ( ( 1) ) ( ) )

2 4

*m*

.

*M* *b* *n* *b* *n* *b k*

Now (5) follows by the fact that *b k*, ( ) =*G* *m*2 |*M*|.

**3. An Upper Bound on SbkECN**

Bonato *et al.* in [11] posed the following conjecture on
( )

*b* *G*
.

**Conjecture 5 **Let be an integer. There is a
positive integer so that for any graph *G* of order

with minimum degree *b*,
2

*b*

*b*

*n*

*b*

*n n*

( ) ( 1)( 1).

*b* *G* *b* *n b*

Since *b*(*Kb*1,*n b* 1) = (*b*1)(*n b* 1), the upper bo-

und would be the best possible if the conjecture were true. They also proved that the conjecture is true for . In this section we provide an upper bound for = 2

*b*

, '(

*b k* *G*)

, where *b*= 2 and 1 *k n*. The proof of the
next theorem is essentially similar to the proof of
Theorem 5 in [11].

**Theorem 6 ***Let G be a graph of order , size *
*and without isolated vertices. Let * * be the number *
*of vertices with degree at least 2. Then for * * and *

*, *

*n*

*n*

*m*

0 > 0

*n*

7

0

1 *k* *n*

2,*k*( ) 2*G* *n k* 9.

*Proof.* The proof is by induction on the size *m* of *G*.
By a tedious and so omitted argument, it follows that

2,*k*( )*G* *k* 5

if . We may therefore assume that

. Suppose that the theorem is true for all graphs *G*
without isolated vertices and size less than . Let *G* be
a graph of order , size and without isolated
vertices. We will prove that

= 7

*n*

8

*n*

*m*

2

*G* *n*

8

*n* *m*

2,*k*( ) *k* 9
for
each 1 *k* *n*_{0}

( )*G*

. We consider four cases.
**Case 1.** = 1.

Let be a vertex of degree 1 and . First s-
uppose . Then the induced subgraph *G u*[

2

*u*
deg( ) =*v*

(

*v N* *u*)
,*v*

1 ] is

*K* . It is straightforward to verify that 2,*k* 2*n k* 9
ence, we assume that *n*9. Let
=

*G* *v*

when *n*= 8. H

*G u*

may

. Then *G* is a graph of rder o *n* 2 7,
size *m*1 and without isolated vertices. By the induc-
tive hypothesis, _{2,}* _{k}*( )

*G* 2(

*n* 2)

*k*9 = 2

*n k* 13. Let

*f*be a 2,

*k*( )

-1 and ( ) =

*G*

cover. Define 1,1}

by *g uv*( ) = ( )

: (

*g E* *G*){ *g e* *f e*

*uv*

if *e E G* ( ) . Obviously, *g* is a S2kEC and so

2,*k*( )*G* 2,*k*( )*G* 1 2*n k* 14 < 2*n k* 9.

r two subcases.

*G u* , (*G u*)
Now suppose

i

deg( )*v*
( )*v*
hypothesis

2. Conside

on ve

**Subcase 1.1** deg 3.

By the induct _{2,}* _{k}*

2(*n* 1) *k* 9 = 2*n k* 11

. Le t *f* b e a 2,*k*

(*G u* ) -cover a
= 1

nd define *g E G*: ( ) { 1,1} by
( )

*g uv* and *g e*( ) = ( )*f e* if *e E* ( )*G* *uv* .
Obviously, *g* is a

( ) 1 2*G* *n k*

S2kEC and so

2,*k*( )*G* 2,*k* 10 < 2*n k* 9.

**Subcase 1.2** deg( ) =*v* 2.

{ }*u* . If = 1 define
( ) =*vw*

*k* , then

Let *w N v* ( )
by

: (*E G*) { 1,1} *g uv*( ) =*g* 1 and ( ) =

*g* *g e*

1

if *e E G* ( )\{ ,*uv vw*}. Obviously, *g* is a S2k

2 9.

EC
of *G* and we ha

2, ( )

ve

( ( )) = 4

*m* *n*

2. By the inductive

*k*

*k*

Let 2*k* . It
h e s is

*G* *g E*

follows that

*G*

0

*n*

h ypo t on *G*{ }*u* , 2,*k*1( { }) 2(*u* *n* 1)
( 1) 9 = 2

*G*

12

*k* *n k* . L e t *f* b e a _{2,}_{k}_{}_{1}
(*G*{ })*u*

( )

*g uv*

-cov
= (*g v*

er. Define *g E*: (*G*) { 1,1} by
)

) = 1

*w* and *g e*( ) = (
ly,

*f e* if *e E* ( ) \*G*

{ ,*uv vw*}. Obvious *g* is a S2kE and so

}) C

3 2 9.

2, ( ) 2, 1( {

**Case 2.** ( ) = 2*G* .

Let be a vertex of degree 2 and . Consider two subcases.

*w* *N w*( ) = { , }*u v*

**Subcase 2.1** . Let be the graph

obtained from by adding an edge

*uv*

. Then
has order , size and at least
( )

*uv E G*

{ }

*G* *w*

1

*n*

*G*

*G* *m*1 *k*1

vertices with degree at least 2. By the inductive hypothesis,

( 1),2( ) 2( 1) ( 1) 9 = 2 12.

*k* *G* *n* *k* *n k*

Let *f* be a _{2,}* _{k}*( )

*G* -cover. Define

*g E G*: ( )

{ 1,1} by *g uw*( ) = ( ) = 1*g vw* and *g e*( ) = ( )*f e* if
. Obviously,

( ) \{*G* *uv vw*, }

*e E* *g* is a S2kEC and so

2, ( ) ( ( )) ( ( )) 3 2 9.

_{k}*G* *g E G* *f E G* *n k*

**Subcase 2.2** . First let both and
have degree 2. Then the induced subgraph

is an isolated triangle. If 1 , then define by

( )

*uv E G*

1,1}

*u*
[{

*G u*

*v*
}]

*w*

, ,*v*

3
*k*
: ( ) {

*f E G*

( ) = ( ) = ( ) = 1 a ( ) = 1 o .

*f uv* *f vw* *f uw* *nd f e* *therwise*

Then

2, ( ) ( ( )) = 6 2 9.

_{k}*G* *f E G* *m* *n k*

Now suppose that *k*4. It is not hard to show that
9

2, ( ) 2

_{k}*G* *n k*
10

*n*

when or 9. Hence, we may

assume that . Let . Then

= 8

*n*

= \

*G* *G* { , , }*u v w* *G*

= 2 is a graph of order , size and has at least vertices with degree at least 2. By the inductive h y p o t h e s i s ,

3

( 3)(

7

2(

*n*

3

*k*

2,

3

*m*

) 3) ( 3) 9

*k* *G* *n* *k* *n*

18

*k* . Let *f* be a 2,(*k*3)( )*G* -cover. Define

by : (

*g E* *G*) { 1,1}

( ) = ( ) = ( ) = 1 a ( ) = ( ) i ( ).

*g uv* *g vw* *g uw* *nd g e* *f e f e E G*

Obviously, *g* is a S2kEC of *G* and

2, ( ) = ( ( )) ( ( )) 3 (2 18) 3.

_{k}*G* *g E G* *f E G* *n k*

Now let . If , define

by and

min{deg( ), deg( )} 3*u* *v*

{ 1,1}

*g uw*( ) = (*g v*

= 1

*k*

= 1 : ( )

*g E G* *w*) *g e*( ) =

1

otherwise. Obviously, *g* is a S2kEC and so

2, ( ) ( ( )) = 4 < 2 9.

_{k}*G* *g E G* *m* *n k*

}

If , then is a graph of order , size and has at least vertices with degree at least 2. By the inductive hypothesis, we have

that

2

*k* *G*=*G*{*w*

1

*n* *m*2 *k*1

2,( 1)( ) 2 12

*k* *G* *n k*

( )

. Let *f* be a

2,( 1)

_{k}_{} *G* -cover. We can obtain a S2kEC *g* of by
assigning for each

*G*

) ( ) = 1

*g e* *e E G* ( )\*E*(*G* and

( ) = ( )

*g e* *f e* for each *e E G* ( )

2,(

( )) = ( ( )) 2 =

. Then we have

1)( ) 2

( *G* *f E G* *k* < 2*n**k*9.

2, ( ) < 2 9

*g E* *G*

Hence, _{k}*G* *n k*

min{deg( ),
g( ) = 2*u*

) { 1,1}

*G* *g uw*( )

= 1

, as desired.

Finally, assume . Let without loss of generality de . If 1 , define

by and

deg( )} = 2

*u* *v*

2
= (*g vw*) = )

(

*g uv*
*k*

: (

*g E*

( )

= 1

*g e* otherwise. Obviously, *g* is a S2kEC and so

2, ( ) ( ( )) = 6 2 9.

*m* *n*

}

*k*
*k* *G* *g E G*

3

*k* = {

If , then *G* *G* *w* is a graph of order
1

*n* , size *m*2

2,( 2)( ) 2( 1) (

and has at least vertices with degree at least 2. By the inductive hypothesis,

2

*k*

) 9 = 2

2 13.

_{k}_{} *G* *n*

*f* 2,( 2)( )

*k* *n**k*

Let be a *k* *G*

}

-cover. Define *g E*: ( )*G*

{ 1,1 by

( ) = ( ) = ( ) = 1 a ( ) = ( )

*g uv* *g vw* *g uw*

i (

*nd g e* *e*

) \{ }.

*f*

*f e E G* *uv*

Obviously, *g* is a S2kEC of *G* and

)) 4

2, ( ) ( ( )) = ( ( 2 9.

_{k}*G* *g E G* *f E* *G* *n**k*

( ) = 3

**Case 3.** *G*

*w*

) { 1,1}

*G* *g uw*( )

= 1

.

Let be a vertex with degree 3. If , define

by if and

= 1

*k*

*u N*

: (

*g E*

( )

= 1 ( )*w*

*g e* otherwise. Obviously, *g* is a S2kEC and so

2, ( ) ( ( )) = 6 < 2 9.

*m* *n*

}

*k* *G* *g E G*
2

*k* = {

*k*

If , then *G* *G* *w* is a graph of order
1

*n* , size *m*3

2,( 1)( ) 2 12

and has at least vertices with degree at least 2. By the inductive hypothesis, we have that

1

*k*

*k* *G* *n k* . Let *f* be a 2,( *k*1)

( )*G* -cover. We can obtain a S2kEC *g* of by
assigning for each

*G*

) ( ) = 1

*g e* *e E G* ( )\*E*(*G* and

( ) = ( )

*g e* *f e* for each *e E G* ( )

2,(

( )) = ( ( )) 3 =

. Then we have

1)( ) 3

( *G* *f E G* *k* *G* 2 *k*9.

2, ( ) 2 9

*g E* *n*

**5. References**

Then *G*has an even cycle by Theorem 6. Let

1 2,

= ( , , )*s*

*C* *v* *v*

= ( )

*G* *G* *C*

*v*
*E*

be an even cycle in *G*. Obviously,
is a graph of order , size *n* *m*| ( ) |*E C*

and has at least vertices with degree at least 2. By the inductive hypothesis,

*k*

2, ( ) 2 9

_{k}*G* *n k* . Let *f* be a

2, ( )

_{k}*G* -cover. Let *vs*1 =*v*1 and define *g E G*: ( )

[1] D. König, “Über trennende knotenpunkte in graphen (nebst anwendungen auf determinanten und matrizen),” Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae, Sectio Scientiarum Mathematicarum [Szeged], Vol. 6, pp. 155–179, 1932– 1934.

{ 1,1} by

[2] R. Rado, “Studien zur kombinatorik,” Mathematische Zeitschrift, German, Vol. 36, pp. 424–470, 1933.

1

( _{}) = ( 1) i *i* = 1, , a ( ) = ( ) f
*i i*

*g v v* *f i* *s nd g e* *f e or*

[3] T. Gallai, “Über extreme Punkt-und Kantenmengen
(German)*,*” Ann. Univ. Sci. Budapest. Eötvös Sect. Math.,
Vol. 2, pp. 133–138, 1959.

( ) \ ( ).

*e E G E C*

Obviously, *g* is a S2kEC and hence _{2,}*k*( ) =*G*

[4] R. Z. Norman and M. O. Rabin, “An algorithm for a minimum cover of a graph,” Proceedings of the American Mathematical Society, Vol. 10, pp. 315–319, 1959.

2, ( ) 2 9.

_{k}*G* *n k* This completes the proof.

**4. Conclusions**

[5] A. Schrijver,“Combinatorial optimization: Polyhedra and
Efficiency,” Springer, Berlin, 2004.
[6] D. B. West, “Introduction to graph theory,” Prentice- Hall, Inc., 2000.

In this paper we initiated the study of the signed
-edge cover numbers for graphs, generalizing the
signed star domination numbers, the signed star *k*-
domination numbers and the signed *b*-edge cover
numbers in graphs. The first lower bound obtained in this
paper for the signed -edge cover number conc-
ludes the existing lower bounds for the other three pa-
rameters. Our upper bound for the signed -edge
cover number also implies the existing upper bound for
the signed *b*-edge cover number. Finally, Theorem 6
inspires us to generalize Conjecture 5.

( , )*b k*

( , )*b k*

( , )*b k*

[7] B. Xu,“On signed edge domination numbers of graphs,” Discrete Mathematics, Vol. 239, pp. 179–189, 2001. [8] C. Wang, “The signed star domination numbers of the

Cartesian product,” Discrete Applied Mathematics, Vol. 155, pp. 1497–1505, 2007.

[9] B. Xu, “Note on edge domination numbers of graphs,” Discrete Mathematics, Vol. 294, pp. 311–316, 2005. [10] B. Xu,“Two classes of edge domination in graphs,” Discr-

ete Applied Mathematics, Vol. 154, pp. 1541–1546, 2006.
**Conjecture 7 **Let be an integer. There is a

positive integer so that for any graph *G* of order
with vertices of degree at least *b*, and for
any integer ,

3
*b*

,

* _{b k}*

*b*

*n*

1

*k*

_{b}

*n n* *n*0

1 *n*0

2

( )*G* *bn k* (*b* 1) .

[11] R. Saei and S. M. Sheikholeslami, “Signed star *k*
-sub-domination numbers in graph,” Discrete Applied
Mathe-matics, Vol. 156, pp. 3066–3070, 2008.