How to construct new super edge-magic graphs from some old ones

How to construct new super edge-magic
graphs from some old ones
E.T. Baskoro1? , I W. Sudarsana2?? and Y.M. Cholily1
2
1
Department of Mathematics
Institut Teknologi Bandung (ITB),
Jalan Ganesa 10 Bandung 40132, Indonesia
{ebaskoro, yus}@dns.math.itb.ac.id
Department of Mathematics, Tadulako University,
Jalan Sukarno-Hatta Palu, Indonesia
[email protected]
Abstract. In this paper, we study the property of super edge-magic
total graphs. We give some further necessary conditions for such graphs.
Based on this condition we provide some algorithms to contruct new
super edge-magic total graphs from some old ones.
Keywords : super, edge-magic total labeling
1
Introduction
All graphs, in this paper, are finite and simple. A general reference
for graph-theoretic ideas can be seen in [4]. For a graph G with
vertex-set V (G) and edge-set E(G) an edge-magic total labeling is a
bijection λ: V (G)∪E(G) → {1, 2, · · · , |V (G)∪E(G)|} with satisfying
the property that there exists an integer k such that
λ(x) + λ(xy) + λ(y) = k,
for any edge xy in G. We call λ(x)+λ(xy)+λ(y) the edge sum of xy,
and k the magic constant of graph G. In particular, if λ(V (G)) =
{1, 2, · · · , |V (G)|} then λ is called super edge-magic total labeling.
A graph is called (super) edge-magic total if it admits any (super)
edge-magic total labeling.
The notion of edge-magic total graphs was introduced and studied by Kotzig and Rosa [5] with a different name as graphs with
magic valuations, while the term of super edge-magic total graphs
was firstly introduced by Enomoto et al. [1]. They showed that a
?
??
Supported by Hibah Bersaing XII DP3M-DIKTI Indonesia, 2004, DIP Number:
004/XXIII/1/–/2004.
Supported by Hibah Pekerti DP3M-DIKTI Indonesia, 2004.
2
E.T. Baskoro, I W. Sudarsana and Y.M. Cholily
star Sn+1 = K1,n is the only complete bipartite graph which is super edge-magic total. They also showed that any odd cycle is super
edge-magic total, but any wheel is not.
Since then, a number of papers have studied super edge-magic
property in graphs. For instances, Figueroa-Centeno et al. [3] and [2]
derived a necessary and sufficient condition for a graph to be super
edge-magic total and they also showed several class of graphs, such
as fans fn ∼
= Pn +K1 with n ≤ 6, ladders Ln ∼
= Pn ×P2 for odd n, and
the generalized prism G ∼
C
×
P
for
odd
m and n ≥ 2, are super
= m
n
edge-magic. They also studied the relationships between super edgemagic labeling with other labelings. However, a conjecture ”Every
tree is super edge-magic total” proposed by Enomoto et al. [1] still
remains open.
In this paper, we study super edge-magic total labelings. We
derive more necessary conditions to be able to know more deeply
the property of such labelings. Based on this condition we give some
algorithms to contruct a new super edge-magic labeling from some
old ones. By using these algorithms we can provide more evidence
to support the correctness of the conjecture proposed by Enomoto
et al.
2
Some further necessary conditions
Several necessary conditions for a graph to be super edge-magic have
been derived by several authors. Enomoto et al. [1] showed that if a
nontrivial graph G is super edge-magic then |E(G)| ≤ 2|V (G)| − 3.
Furthermore, Figueroa-Centeno et al. [3] provide a neccessary and
sufficient condition for a graph being super edge-magic as in the
following lemma.
Lemma 1. A (p, q)-graph G is super edge-magic if and only if there
exists a bijective function f : V (G) → {1, 2, · · · , p} such that the set
S = {f (u) + f (v) : uv ∈ E(G)}
consists of q consecutive integers. In such a case, f extends to a super
edge-magic labeling of G with the magic constant k = p+q +s, where
s = min(S) and
S = {f (u) + f (v) : uv ∈ E(G)}
= {k − (p + 1), k − (p + 2), · · · , k − (p + q)}.
How to construct new super edge-magic graphs
3
Further, in order to know what possible values of k 0 s for graph
G to be super, we add the following neccessary conditions.
Lemma 2. Let a (p, q)-graph G be super edge-magic total. Then,
the magic constant k of G satisfies p + q + 3 ≤ k ≤ 3p.
Proof. Since G is super edge-magic total then the vertices of G receive
labels 1, 2, · · · , p and the edges receive p + 1, p + 2, · · · , p + q
so that by Lemma 1 S = {f (u) + f (v) : uv ∈ E(G)} consists of
consecutive integers a, a + 1, · · · , a + q − 1 for some positive integer
a. The smallest possible magic constant of G obtained if a = 3.
In this case the vertices of G with labels 1 and 2 are adjacent and
the magic constant for this case must be k = (a + q − 1) + (p + 1)
= p + q + 3. If the vertices of labels p − 1 and p are adjacent in G
then we obtain the biggest possible magic constant of G , namely
k = (p−1)+p+(p+1) = 3p. Therefore we obtain p+q +3 ≤ k ≤ 3p.
u
t
The lower and upper bounds in Lemma 2 are tight, since the
super edge-magic labelings λ1 and λ2 on a star Sn of n vertices
shown in Fig. 1 have the magic constant 2n + 2 and 3n, respectively.
2
3
n
2n-2
2n-1
λ1
1
1
n+1
2
n-1
2n-2
2n-1
n
n+1
λ2
Fig. 1. A star graph achieves the lower and upper bounds of k.
Corollary 1 If k is the magic constant of a tree with p vertices then
2p + 2 ≤ k ≤ 3p. Furthermore, the magic constant of a (p, q)-graph
G with c components ranges between 2p − c + 3 to 3p.
4
E.T. Baskoro, I W. Sudarsana and Y.M. Cholily
Proof. The first statement holds, since in any tree the number of
edges is one less than the number of vertices. If a (p, q)-graph G has
c components then in each component Gi (i = 1, 2, · · · , c) we have
E(Gi ) ≥ V (Gi ) − 1. Thus, |E(G)| ≥ |V (G)| − c and by Lemma 2 it
implies that 2p − c + 3 ≤ k ≤ 3p.
u
t
3
Duality in super edge-magic labeling
Given any edge-magic total labeling λ on a (p, q)-graph G, Wallis et
al. [6] define the dual labeling λ0 of labeling λ as follows.
λ0 (vi ) = M − λ(vi ), ∀vi ∈ V (G), and
λ0 (x) = M − λ(x), ∀x ∈ E(G),
where M = p + q + 1.
It is easy to see that if λ is edge-magic total with the magic
constant k then λ0 is edge-magic total with the magic constant k 0 =
3M − k. It is also easy to see that if λ is super edge-magic total then
λ0 is no longer super edge-magic total.
In the next theorem, we introduce another dual property which
preserve the superness of edge-magic total labelings.
Theorem 1. Let a (p, q)-graph G be super edge-magic total. Let λ
be a super edge-magic total labeling of G with the magic constant k.
Then, the labeling λ0 defined:
λ0 (vi ) = p + 1 − λ(vi ), ∀vi ∈ V (G), and
λ0 (x) = 2p + q + 1 − λ(x), ∀x ∈ E(G)
is a super edge-magic total labeling with the magic constant k 0 =
4p + q + 3 − k.
Proof. Let uv ∈ E(G). Then, λ0 (u) + λ0 (uv) + λ0 (v)
= (p + 1 − λ(u)) + (2p + q + 1 − λ(uv)) + (p + 1 − λ(v))
= 4p + q + 3 − (λ(u) + λ(uv) + λ(v))
= 4p + q + 3 − k a constant.
Therefore, λ0 is a super edge-magic total labeling of G with magic
constant k 0 = 4p + q + 3 − k.
u
t
The labeling λ0 in Theorem 1 is called the dual super labeling of
λ on G.
How to construct new super edge-magic graphs
4
5
Construction of new labelings
In this section, we give algorithms to construct new super edge-magic
graphs by extending the old ones.
Theorem 2. From any super edge-magic (p, q)-graph G with the
magic constant k, we can construct a new super edge-magic total
graph from G by adding one pendant incident to vertex x of G whose
label k − 2p − 1. The magic constant of the new graph is k 0 = k + 2.
Proof. In the new graph, define a labeling in the following. Preserve
all vertices labels of G. Increase the labels of all edges (except the
new one) by 2. Label the new vertex and edge by p + 1 and p + 2
respectively. It can be verified that the resulting labeling on the
new graph is super edge-magic total labeling with magic constant
k = k + 2. Since 2p + 2 ≤ k ≤ 3p (by Lemma 2), the proposition
holds for any value of k.
u
t
Theorem 3. Let a (p, q)-graph G be super edge-magic total with the
magic constant k and k ≥ 2p + 3. Then, a new graph formed from
G by adding exactly two pendants incident to two distinct vertices x
and y of G whose labels k − 2p and k − 2p − 2 respectively is super
edge-magic total with the magic constant k 0 = k + 4.
Proof. In the new graph denote by u and v the new vertices adjacent
to x and y, respectively. Then, define a labeling in the new graph as
follows. Preserve labels of all the vertices of G. Add all edge labels
(except the new ones) by 4. Label vertices x and y by p+1 and p+2,
respectively and label two new edges xu and yv by p + 3 and p + 4,
respectively.
For the new edges, clearly we have the edge sum of each is k + 4.
Since each label of old edge increased by 4 then we get the edge sum
of each old edge is also k + 4. Therefore, the new graph is super
edge-magic total labeling. This process works only if k − 2p − 2 ≥ 1.
This implies that k ≥ 2p + 3.
u
t
Theorem 4. Let a (p, q)-graph G be super edge-magic total with the
magic constant k and k ≥ 2p + 3. Then, a new graph formed from G
by adding exactly three pendants incident to three distinct vertices x ,
y and z of G whose labels k −2p, k −2p−1 and k −2p−2 respectively
is super edge-magic total with the magic constant k 0 = k + 6.
Proof. In the new graph, define a labeling as follows. Preserve all
vertex labels of G in the new graph. Increase all edge labels (except
6
E.T. Baskoro, I W. Sudarsana and Y.M. Cholily
the new ones) by 6 in the new graph. Label the three new vertices
u which adjacent to x, y and z by using the second row of either
matrix A or B. Label the corresponding new edge e by using the
third row from A or B.



x
y
z
x
y
z
A = u : p + 1 p + 3 p + 2 , B = u : p + 2 p + 1 p + 3
e: p+4 p+6 p+5
e: p+5 p+4 p+6

For the new edges, clearly we have the edge sum of each is k + 6
(from the above matrix). Since each label of old edge increased by 6
then we get the edge sum of each old edge in the new graph is k + 6.
Therefore, the new graph is super edge-magic total. Note that this
process works only if k −2p−2 ≥ 1. This implies that k ≥ 2p+3. u
t
Alternatively, we have the following theorem for adding three
pendants.
Theorem 5. Let a (p, q)-graph G be super edge-magic total with the
magic constant k and k ≥ 2p + 4. Then, a new graph formed from G
by adding exactly three pendants incident to three distinct vertices x
, y and z of G whose labels k − 2p + 1, k − 2p − 1 and k − 2p − 3
respectively is super edge-magic total with the magic constant k 0 =
k + 6.
Proof. The proof is similar with the one of Theorem 4 by using the
following matrix C.
u
t


x
y
z
C = u : p + 1 p + 2 p + 3
e: p+4 p+5 p+6
Theorem 6. Let p be an odd integer. Let a (p, q)-graph G be super
edge-magic total with the magic constant k = (5p + 3)/2. Then, a
new graph formed from G by adding exactly p pendants incident to all
vertices of G is also super edge-magic total with the magic constant
k = (9p + 3)/2.
Proof. In the new graph, define a labeling as follows. Preserve all
vertex labels of G in the new graph. Increase all edge labels (except
the new ones) by 2p in the new graph. Label each new vertex u which
adjacent to the old vertex v by using the second row of the following
matrix. Label the corresponding new edge e = vu by using the third
row.
How to construct new super edge-magic graphs
7


p+1
p+3
v:
1
2
· · · p−1
·
·
·
p
−
1
p
2
2
2
u : 3p+3 3p+5 · · · 2p p + 1 p + 2 · · · 3p−1 3p+1 
2
2
2
2
e : 3p − 1 3p − 3 · · · 2p + 2 3p 3p − 2 · · · 2p + 3 2p + 1
For the new edges, clearly we have the edge sum of each is 9p+3
2
(from the above matrix). Since each label of old edge increased by
2p then we get the edge sum of each old edge in the new graph is
k + 2p = 9p+3
. Therefore, the new graph is super edge-magic total.
2
u
t
Note that the (extension) construction method in Theorem 6 only
works for a super edge-magic total graph with the magic constant
k = (5p + 3)/2. There are several graphs of p vertices known to have
the magic constant (5p + 3)/2, such as odd cycles and paths with
odd number of vertices.
Let C(n, s) be a graph contructed from path Pn of n vertices
by adding s pendants to each vertex of Pn . Let us call C(n, s) by
a caterpillar with s legs. We know that C(n, s) is super edge-magic
total [5]. The following corollary shows one way how to label this
graph so that super edge-magic total.
Corollary 2 For odd n and s ≥ 1, the graph C(n, s) is super edgemagic total.
Proof. Take a super edge-magic total labeling for path Pn , for odd n,
with the magic constant (5n + 3)/2, namely label the vertices in the
odd positions of Pn from left to right consecutively by 1, 2, · · · , (n +
1)/2; And then label the even positions from left to right consecutively by (n + 3)/2, (n + 5)/2, · · · , n; Next, it is easy to label all the
edges of Pn so that we have a super edge-magic total labeling.
Apply Theorem 6 to Pn . Denote the resulting graph by C(n, 1).
For i = 1, 2, · · · , s − 1, apply Theorem 6 repeatedly to graph C(n, i)
and denote the resulting graph by C(n, i + 1). In the final result, we
have a super edge-magic labeling for C(n, s).
u
t
Let Tp be a tree of p vertices, for p ≥ 3. For h ≥ 1 we denote
by Tn + Ah a graph which is obtained by adding h pendants to one
vertex of tree Tp . Then, we have:
Theorem 7. From any super edge-magic total tree Tp with the magic
constant k = 2p + 1 + s, for some s ∈ {1, 2, · · · , p}, we can construct
a new super edge-magic total tree Tp + Ah .
8
E.T. Baskoro, I W. Sudarsana and Y.M. Cholily
Proof. Apply Theorem 2 to Tp h times by attaching a new pendant
each to vertex whose label k − 2p − 1.
By using Theorem 7 we can obtain a class of trees which is super edge-magic total from just one super edge-magic total tree. For
instances, the tree in Fig. 2(b) is obtained by applying Theorem 7
to the tree in (a). This theorem provides more facts to support the
correctness of the Conjecture proposed by Enomoto et al. [1].
1
9
8
6
10
2
4
7
5
11
3
(a)
1
2h+11
2h+10
6
2
2h+12
4
2h+9
5
2h+13
(b)
3
h+8
2h+8
2h+7
7
h+7
8
Fig. 2. The tree in (b) is formed from the tree in (a) by using Theorem 7.
References
1. H. Enomoto, A.S. Llado, T. Nakamigawa and G. Ringel: Super edge-magic graphs,
SUT J. Math. 2 (1998), 105-109.
2. R.M. Figueroa-Centeno, R. Ichishima and F.A. Muntaner-Batle, On super edgemagic graphs, Ars Combin., 64 (2002) 81-95.
3. R.M. Figueroa-Centeno, R. Ichishima and F.A. Muntaner-Batle, The place of super
edge-magic labelings among other classes of labelings, Discrete Math., 231 (2001)
153-168.
4. N. Hartsfield and G. Ringel: Pearls in Graph Theory (Academic Press, 1990).
5. A. Kotzig and A. Rosa: Magic valuations of finite graphs, Canad. Math. Bull. 13
(1970), 451-461.
6. W. D. Wallis, E. T. Baskoro, M. Miller and Slamin: Edge-magic total labelings,
Australasian Journal of Combinatorics 22 (2000) 177-190.