View of Expanding Super Edge-Magic Graphs∗

Expanding Super Edge-Magic Graphs

^∗

E. T. Baskoro¹ & Y. M. Cholily^1,2

1 Department of Mathematics, Institut Teknologi Bandung Jl. Ganesa 10 Bandung 40132, Indonesia

Emails : {ebaskoro,yus}@dns.math.itb.ac.id

2 Department of Mathematics, Universitas Muhammadiyah Malang Jl. Tlogomas 246 Malang 65144, Indonesia

Email : yus@umm.ac.id

Abstract. For a graph G, with the vertex set V(G) and the edge set E(G) an edge- magic total labeling is a bijection f from V to the set of integers

with the property that ) ( ) (G ∪E G , f

, 2 , 1

{ " |V(G)|+|E(G)|} (u)+ f(v)+ f(uv)=k for each uv∈E(G)and for a fixed integer k. An edge-magic total labeling f is called super edge-magic total labeling if f( GV( ))={1,2," |,V(G)|} and

, 1

| ) (

{| +

= V G |V(

)) ( ( GE

f G)|+ "2, |,V(G)|+|E(G)|}. In this paper we construct the expanded super edge-magic total graphs from cycles C , generalized Petersen graphs and generalized prisms.

n

Keywords: Edge-magic; super edge-magic; magic-sum.

1 Introduction

All graphs considered here are finite, undirected and simple. As usual, the vertex set and edge set will be denoted V and respectively. The symbol

)

(G E(G),

A will be denote the cardinality of the set A. Other terminologies or notations not defined here can be found in [2,7,15].

Edge-magic total labelings were introduced by Kotzig and Rosa [8] as follow.

An edge-magic total labeling on is a bijection from VG f (G)∪E(G) onto

|}

) (

|

| ) (

|, , 2 , 1

{ " V G + E G with the property that, given any edge uv, k

uv f v f u

f( )+ ( )+ ( )=

for some constan k. It will be convenient to call f(u)+ f(v)+ f(uv) the edge sum of uv and k the magic sum of f. A graph is called edge-magic total if it admits any edge-magic total labeling.

∗ This work was supported by Hibah Bersaing XII DP3M-DIKTI, DIP Number : 004/XXIII/1/--/2004.

Kotzig and Rosa [9] showed that no complete graph with is edge- magic total and neither is and edge-magic total labelings for and

for all feasible values of k, are described in [14].

Kn n>6 K3 4,

K , K₅

K6

In [8] it is proved that every cycle every caterpillar (a graph derived from a path by hanging any number of pendant vertices from vertices of the path) and every complete bipartite graph (for any and ) are edge-magic total.

n, C

n ,

Km m n

Wallis et.al. [14] showed that all paths and all n-suns (a cycle C with an additional edge terminating in a vertex of degree 1 attached to each vertex of the cycle) are edge-magic total. It was shown in [16] that the Cartesian product

admits an edge-magic total labeling for odd n.

Pn _n

m

n P

C ×

It is conjectured that all trees are edge-magic total [8] and all wheels W are edge-magic total whenever n°3 (mod 4) [4]. Enomoto et.al. [4] showed that the conjectures are true for all trees with less than 16 vertices and wheels W for Philips et.al. [12] solved the conjecture partially by showing that a wheel

n

n

.

≤30 n

n,

W n≡0 or 1 6

(mod 4), is edge-magic total. Slamin et.al [13] showed that for n≡ (mod 8) every wheel W has an edge-magic total labeling. _n

An edge-magic total labeling f is called super edge-magic total if and

|}

) (

|, , 2 , 1 { )) (

(V G V G

f = " f(E(G))={|V(G)|+1,| V(G)|+2,",| V(G)|+

n

Km_,

1

Enomoto et.al. [4] proved that the complete bipartite graphs is super edge-magic total if and only if

|}.

) (

|E G

=

m or n=1. 2 , 1

They also proved the complete graphs K_n is super edge-magic if and only if n= or 3.

In this paper we will construct the super edge-magic total graphs by hanging any number of pendant vertices from vertices of the cycles, generalized prisms and generalized Petersen graphs.

2 Results

For and n≥3 p≥1 we denote by C_n+A_p a graph which is obtained by adding p vertices and p edges to one vertex of cycles C (say ). The vertex set and the edge set of are V

n

1 : v_i

v1

}

p n +A

Cn (C_n+A_p)={ ≤i≤ ∪{u_j:1≤ j≤ p} and }.

1 { }

1 {v_nv₁} v₁u_j: j 1

: i {v_iv_{i 1} )

( n

E C_n+A_p = ₊ ≤ ≤ − ∪ ∪ ≤ ≤ p

Let be a graph derived from a cycle C by hanging p pendant vertices from all vertices of the cycle. Let us denote the vertex set of

y V sun p n, )− (

sun p)−

( b

n, 1 :

,

≥3 n

1 , n i

n, ((n,p)−sun)={v_i:1≤i≤n}∪{u_i,j ≤ ≤ ≤ j≤ p} 1 : }

1 1

: { ) ) ,

((n p sun v_iv_{i 1} i n ∪ _iu_i,j ≤i and the edge set by E − = ₊ ≤ ≤ − { vv_{n 1}}∪{v ≤n,

).

1 ( ,

((

| | ) ) , ((

| +

Observe that }.

p j≤

1≤ V n p −sun = E n p)− ) |=n p

p

n A

C +

n. C

sun The

cycle is super edge-magic total if and only if n is odd (see [4]). Now, we shall investigate super edge-magic total labelings for graphs of and

which are expanded from a cycle ,

3 ,n≥ C_n

sun p)− n, (

Define a vertex labeling and an edge labeling f₁ f₂ of C_n+A_p as follows,





= ⁺

even, is if

odd, is ) if

(

2 1 2

i v i

f _i

i n i

j n u

f₁( _j)= + for 1≤ j≤ p, i p n v

v

f₂( _{i i+1})=2( + )+1− for 1≤i≤n−1, ,

1 2 ) ( ₁

2 v v =n+ p+ f _n

for 1 j p n u v

f₂( _{1 j})= +2 +1− ≤ j≤ p.

Theorem 1. If n is odd, n≥3 and p≥1, then graph C_n+A_p is super edge- magic total.

Proof. It is easy to verify that the values of are f₁ 1, "2, ,n+p and the values of f₂ are n+ p+1 ,n+p+2 ," ,2n+2p and furthermore the common edge sum is k=2p+⁵ⁿ₂⁺³. ;

Theorem 2. If n is odd, n≥3 and p≥1,then graph (n,p)−sun is super edge- magic total.

Proof. Label the vertices and the edges of (n,p)−sunin the following way.

) ( )

( ₁

3 vi f vi

f = for 1≤i≤n, for 1

1 j p,

) ( _1,

3 u = nj+

f _j ≤ ≤

for i j

n u

f₃( _i,j)= ( +1)+2− 2≤i≤n and 1≤ j≤ p, i

p n v

v

f₄( _{i i+1})=2 ( +1)+1− for 1≤i≤n,

, 1 2

) ( ₁

4 v v = np+n+ f _n







≤

≤

−

≤

≤ +

− +

≤

≤

≤

≤ +

− +

≤

≤

=

− +

=

−

−

−

. 1

and 1 2

even, is if )

1 ( 2

, 1

and 3 odd, is if )

1 ( 2

, 1

and 1 if )

1 ( 2 ) (

2 21 1 ,

4

p j n

i i

nj p

n

p j n

i i

j n np

p j i

nj p

n u

v f

n i j i

i i

We can see that the vertices of n,p)−sun (

( are labeled by values 1 and the edges are labeled by

, , 2 , "

) 1 (p+

n n p+1)+1, n(p+1)+2," ,2n(p+1).

Furthermore, all edges have the same magic number k =2 pn( +1)+ ⁿ₂⁺³. ; A generalized Petersen graph P( mn, ), n≥3 and ^1≤m≤

 

ⁿ₂⁻¹

i, 1 i n,

, consists of an outer n-cycle v₁,v₂,",v_n a set of n spokes v_iz ≤ ≤ and inner edges

,

1 with indices taken modulo n.

m,

i iz

z ₊ ≤i≤n

For n≥5, m=2 and p≥1,we denote by P(n,2)+A_p

} 1

: j p

for a graph which is obtained by adding p vertices and p edges to one vertex of say Hence,

), 2 , (n

P v₁.

= ) V( +

(P(n,2) A_p

V P(n,2)) ∪ {u_j ≤ ≤ and E( nP( ,2)+A_p)= ))

2 , (n (P

E ∪ {v₁u_j:1≤ j≤ p}.

Let be a graph derived from , by hanging p pendant vertices from all vertices

) , 2 , (n p

P P(n,2), n≥5

i,

v 1≤i≤n o P(n 1 : {u_i,j

f Then the vertex set of is V

).

2 ,

1 , n i )

, 2 , p (n

P (P(n,2,p)) V(P(n,2))∪ ≤ ≤ ≤ j≤ p} :

{v_iu_i,j

=

}.

( ( ))

,p E P n p

and the edge set is E( nP( ,2 = ,2))∪ 1≤i≤n ,1 j≤ ≤

In [11] it is proved that generalized Petersen graphs are edge-magic total. Fukuchi [6] showed that are super edge-magic total.

) 2 , (n P )

2 , (n P

Theorem 3. If n is odd, n≥5 and p≥1, then the graph P(n,2)+A_p has a super edge-magic total labeling.

Proof. Consider a bijection, f₅:V(P(n,2)+A_p)→{1 ,2," ,2n+p} where,







≤

≤

−

≤

≤

= +₊

, 1 odd, is if

, 1 2

even, is ) if

(

2

3 2

5 i i n

n i i

v n

f _{n i}

i i











≡

≡

≡

≡

=

+

− +

− +

− +

−

4), (mod 0 if

4), (mod 3 if

4), (mod 2 if

4), (mod 1 if )

(

4 4 4 4 4 3 4 4 2 4 4 5

i i i i z

f

i n

i n

i n

i n

i

j n u

f₅( _j)= 2 + for 1≤ j≤ p.

We can observe that under the labeling f₅, {f₅(v_i)+ f₅(v_i+1): 1≤i≤n}= }

1 :

{⁵ⁿ₂⁺¹+i ≤i≤n and {f₅(z_i)+ f₅(z_i+2):1≤i≤n}={ⁿ₂⁺¹+i: 1≤i≤n} with indices taken modulo n. Moreover, {f₅(v_i)+ f₅(z_i):1≤i≤n}={³ⁿ⁺₂¹+i:1≤i≤n} and j≤p}={⁷ⁿ₂⁺¹+ j: ≤ p}.

}

n ∪{f₅(z_i) f₅( :1 i n} 1 j≤

2) z_i 1

: ) ( ) (

{f₅ v₁ + f₅ u_j ≤ 1 : ) (

) f₅ v ₁ i v_i + _i+ ≤ ≤

The elements of the set (

{f₅ + ₊ ≤ ≤ ∪ { f₅(v_i)+ f₅(z_i):

}

≤n

≤ ∪{f₅(v₁) f₅(

1 i + u_j):1≤ j≤ p} form an arithmetic sequence ⁿ₂⁺¹+1, ,

+2

21 +

n ",⁷ⁿ₂⁺¹, ⁷ⁿ₂⁺¹+1,",⁷ⁿ₂⁺¹+ p. ,

+1

p 2 +2,",

We are able to arrange the values 2n+ n+ p 5n 2+ p to the edges of P(n,2)+A_p

) 2 , ( ( nP E xy

in such way that the resulting labeling is total and every edge ∈ +A_p), f₅(x)+

. 2 )

(

) f₅ xy ¹¹₂ ³ p

y + = ⁿ⁺ +

5(

f Thus we arrive at the desired result. ;

Theorem 4. If n is odd, n and then the graph has a super edge-magic total labeling.

≥5 p≥1, P(n,2,p)

Proof. Define a bijection, f₆:V(P(n,2,p))→{1 ,2," ,n(p+2)} as follows, )

( )

( ₅

6 vi f vi

f = and f₆(z_i)= f₅(z_i) for 1≤i≤n, for 1

1 j p,

) 1 ( ) ( _1,

6 u =n j+ +

f _j ≤ ≤

i j

n u

f₆( _i,j)= ( +2)+2− for 2≤i≤n and 1≤ j≤ p.

We can see that under the vertex labeling f₆ the values f₆(x)+ f₆(y) of all edges constitute an arithmetic sequence xy∈E(P(n,2,p)) ⁿ₂⁺¹+1, ⁿ₂⁺¹+2 ", ,

21,

7n+ ⁷ⁿ₂⁺¹+1,", ⁷ⁿ₂⁺¹+np. If we complete the edge labeling with the consecutive values in the set {n(p+2)+1 ,n(p+2)+2 ,n(p+2)+3 " , , 5n+

= +

}

2np then we can obtain total labeling where f₆(x)+ f₆(y) f₆(xy) np

+2

n 2 3

11 + for every edge xy∈E(P(n,2,p)). ;

In the sequel we shall consider a graph of a generalized prism which can be defined as the Cartesian product C_n×P_m of a cycle on n vertices with a path on m vertices.

Let }V(C_n×P_m)={v_i,k:1≤i≤n and 1≤k≤m

m)

×P {v_i,kv_{i 1,k} :1 i n and 1 k m}

be the vertex set and (C_n

E = ₊ ≤ ≤ ≤ ≤ ∪{v_i,kv_i,k+1:1≤i≤n and ,

≥3

n m≥2

Ap

}

−1

≤ m

, C_n P_m

1≤ k

≥1 p

be the edge set, where i is taken modulo n. For and we will consider a graph ( × )+

+

m)

P

∑

= n

i ip

A

1 m

n P

C

(respectively a graph ) which is obtained by adding p vertices and p edges to one vertex of

× (C_n

× , say (respectively to all vertices f ). Thus V

v_1,m

Pm ((C_n P_m) A_p) V(C_n P_m)

,m,

vi ≤i≤n },

1

: j p

u_j

1 o

Cn× × + = × ∪{ ≤ ≤

}, 1

, 1

: { ) (

) )

(( _,

1

p j n i u P C V A P

C

V ⁿ _{n m i j}

i ip m

n× +

∑

= × ∪ ≤ ≤ ≤ ≤

=

= +

× ) )

((C_n P_m A_p

E E(C_n×P_m)∪{v_1,mu_j:1≤ j≤p}, and }.

1 , 1

: {

) (

) )

(( _{, ,}

1

p j n i u

v P C E A P

C

E ⁿ _{n m im i j}

i ip m

n× +

∑

= × ∪ ≤ ≤ ≤ ≤

=

Figueroa-Centeno et.al. [5] showe that the generalized prism C_n×P_m is super edge-magic if n is odd and m≥2.

The next two theorems show super edge-magic total labelings of graphs and

p m

n P A

C × )+

( ( ) .

∑

1

=

+

× ⁿ

i ip m

n P A

C

Theorem 5. If n is odd, n≥3, m≥2 and p≥1, then the graph (C_n×P_m)+A_p has a super edge-magic total labeling.

Proof. If m is even, m≥2, 1≤k≤m, 1≤i≤n, then we construct a vertex labeling f₇ in the following way,











+

− + +

+

−

= ₋^{− +}

+

even, is and even is if )

1 (

even, is and odd is if

odd, is and even is if

odd, is and odd is if )

1 ( ) (

2 2

2 1 21 ,

7

k i

k n

k i

nk

k i

nk

k i

k n v

f

i n i

n i

i

k i

j mn u

f₇( _j)= + for 1≤ j≤ p.

If m is odd, m≥3,1≤k≤m,1≤i≤n, then we define a vertex labeling as follows,

f8















+

− +

−

=

− +

− +

=

− +

− +

even, is and even is if )

1 (

even, is and odd is if )

1 (

even, is and 1 if

odd, is and even is if )

1 (

odd, is and odd is if )

1 ( )

(

2 1 21 2

2

, 8

k i

k n

k i

k n

k i

nk

k i

k n

k i

k n

v f

i n i i

i n

k i

j mn u

f₈( _j)= + for 1≤ j≤p.

It is easy to verify that for each edge xy∈E((C_n×P_m)+A_p) the values and form an arithmetic sequence

) ( )

( ₇

7 x f y

f + f₈(x)+ f₈(y) ⁿ₂⁺¹+1, ⁿ₂⁺¹+2, ,

" 2mn−ⁿ₂⁻¹, mn2 −ⁿ₂⁻³ ,",2mn−ⁿ_p⁻¹+p.

Let f₉ be a bijection from E((C_n×P_m)+A_p) onto {1 ,2 ," ,2nm−n+p}.

mn f₉+

We can combine the vertex labeling (or ) and the edge labeling

such that the resulting labeling is total and the edge sum for each edge is equal to

f7 f₈ +p

) ) ((C_n P_m A_p

xy∈E × + 3mn+³⁻₂ⁿ+2 p. ;

Theorem 6. If n is odd, and then the graph has a super edge-magic total labeling.

,

≥3

n m≥2, p≥1, (C_n×P_m)+

∑

= n

i ip

A

1

Proof. Define vertex labeling f₁₀and f₁₁ such that : if m is even, 1

) ( )

( _{, 7 ,}

10 vik f vik

f = ≤k≤m, 1≤i≤n,

if m is odd, 1 )

( )

( _{, 8 ,}

11 vik f vik

f = ≤k≤m, 1≤i≤n,

1 ) 1 (

) ( )

( _{1, 11 1,}

10 u = f u =n m+ j− +

f _{j j} for 1≤ j≤p,

2 ) ( ) ( )

( _{, 11 ,}

10 u = f u =n m+ j −i+

f _{i j i j} for 2≤i≤n and 1≤ j≤p.

We can see that vertices of are labeled by values 1, 2, 3,

and for all edges and

constitute an arithmetic sequence

∑

=

+

× ⁿ

i ip m

n P A

C

1

) (

) ( y )

( ,n m+p

"

11}

, 10

∈{ t

) (x f

f_t + _t

∑

=

+

×

∈ ⁿ

i ip m

n P A

C xy

1

) (

, , 2 , 1 ₂¹

21+ ⁺ + "

+ n

n 2mn −

− +

21 n np .

We can complete the edge labeling of

∑

with values in the set

=

+

× ⁿ

i ip m

n P A

C

1

) (

)}

1 2 ,

2 ) ( , 1 ) (

{n m+p + n m+ p + ",n(3m+ p− consecutively such that the common edge sum is k= mn3 +2pn−ⁿ₂⁻³.

∑

=

+

× ⁿ

i ip m

n P A

C

1

) (

Thus the total labeling of is super edge-magic and the theorem is proved. ;

References

1. Bača, M., Consecutive-magic labeling of generalized Petersen graphs, Utilitas Math. 58 (2000), pp. 237-241.

2. Bača, M., MacDogall, J. A., Miller, M., Slamin & Wallis, W. D., Survey of certain valuations of graphs, Discussiones Math. Graph Theory 20 (2000), pp. 219-229.

3. Bača, M., Lin, Y., Miler, M. & Simanjuntak, R., New constructions of magic and antimagic graphs labelings, Utilitas Math. 60 (2001), pp. 229- 239.

4. Enomoto, H., Lladó, A. S., Nakamigawa, T. & Ringel, G., Super edge- magic graphs, SUT J. Math. Vol. 34 (1998), pp. 105-109.

5. Figueroa-Centeno, R. M., Ichishima, R. & Muntaner-Batle, F. A., The place of super edge-magic labelings among other classes of labelings, Discrete Math. 231 (2001), pp. 153-168.

6. Fukuchi, Y., Edge-magic labelings of generalized Petersen graphs P(n,2), Ars Combin. 59 (2001), pp. 253-257.

7. Hartsfield, N. & Ringel, G., Pearls in Graph Theory, Academic Press, New York, 2^nd Edition, 2001.

8. Kotzig, A. & Rosa, A., Magic valuations of finite graphs, Canad. Math.

Bull. 13 (1970), pp. 451-461.

9. Kotzig, A. & Rosa, A, Magic valuations of complete graphs, Publ. CRM 175 (1972).

10. Miller, M. & Bača, M., Antimagic valuations of generalized Petersen graphs, Australasian J. Combin. 22 (2000), pp. 135-139.

11. Ngurah, A. A. G. & Baskoro, E. T., On magic and antimagic total labeling of generalized Petersen graphs, Utilitas Math. 63 (2003), pp. 97- 107.

12. Phillips, N. C. K., Rees, R. S. & Wallis, W. D., Edge-magic total labeling of wheels, Bull. ICA 31 (2001), pp. 21-30.

13. Slamin, Bača, M., Lin, Y., Miller, M. & Simanjuntak, R., Edge-magic total labelings of wheels, fans and frienship graphs, Bull. ICA 35 (2002), pp. 89-98.

14. Wallis, W.D., Baskoro, E. T., Miller, M. & Slamin, Edge-magic total labelings, Australasian J. Combin. 22 (2000), pp. 177-190.

15. Wallis, W. D., Magic Graphs, Birkhäuser, Boston-Basel-Berlin, 2001.

16. Wijaya, K. & Baskoro, E. T., Edge-magic labelings of a product of two graphs, Proc. Seminar MIPA, ITB Bandung (2000), pp. 140-144.