Signed (b,k) Edge Covers in Graphs

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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( )

E

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

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

(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.

Figure

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References