# Signed (b,k) Edge Covers in Graphs

## Full text

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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 Khodkar2, R. Saei3, S. M. Sheikholeslami*3

1Department of Mathematics, University of Mazandaran Babolsar, I.R., Iran 2Department of Mathematics, University of West Georgia, Carrollton, USA 3Department 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 vG 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

EE G( )

### 

and whose

edge 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 eG 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 VG)

( )

e EG 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

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

(2)

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

= { | ( ) } Vv V f v

= 1

b k=

, ,

and .

= { | ( ) = 1} M e E f e 

= { | ( ) < } Vv 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 Gb

( , )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'( ) 2G 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 n0 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 ddn d1d2dn

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 vb { , ,v

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

1 }

jvjk . Define f E G: ( ){0,1} by f e ( ( ) 1) / 2g 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

(3)

( )

( ( 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 vj v jk

k

ji ji b k i

k

ji ji 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 Gm, 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

nm

= ( )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 km 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( )vb} )

and let be a

f

, (

b k G

 -cover. Since for each , ,

it follows that

v Vf v( )b

deg( ) | ( ) |

2

v b

ME 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

(4)

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( ) 2G 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

nm

2,k( ) k 9     for each 1 k n0

( )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 Nu) ,v

1 ] is

K . It is straightforward to verify that  2,k 2n 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 = 2n 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 2n k 14 < 2n 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 = 2n 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 ( )Guv . Obviously, g is a

( ) 1 2G n k

S2kEC and so

2,k( )G 2,k 10 < 2n 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 2k . It h e s is

G g E

follows that

G

0

n

h ypo t on G{ }u , 2,k1( { }) 2(un 1) ( 1) 9 = 2

G

12

   

k n k . L e t f b e a 2,k1 (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( {

(5)

Case 2. ( ) = 2G .

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.

kG  n   k n k 

Let f be a 2,k( )G -cover. Define g E G: ( )

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

( ) \{G uv vw, }

e E g is a S2kEC and so

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

k Gg E Gf 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 Gf E G  m n k 

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

2, ( ) 2

k Gn 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

 kG  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 Gf E G   n k  

Now let . If , define

by and

min{deg( ), deg( )} 3u 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 Gg E Gm 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

 kG  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   < 2nk9.

2, ( ) < 2 9

g E G

Hence, k G n k 

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

) { 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 GGw 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)( )

knk

Let be a  kG

}

-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 Gg E G f E G   nk

( ) = 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 GGw 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

 kG  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   kG  2 k9.

2, ( ) 2 9

g E n

(6)

### 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 =v1 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 n0

1 n0

2

( )Gbn 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.

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