Super edge-magic total labeling of a tree
K. Ali1 , M. Hussain1 , A. Razzaq1
Department of Mathematics,
COMSATS Institute of Information Technology,
Lahore, Pakistan.
{akashifali, mhmaths, asimrzq}@gmail.com
Abstract. An edge-magic total labeling of a graph G is a one-toone map λ from V (G) ∪ E(G) onto the integers {1, 2, · · · , |V (G) ∪
E(G)|} with the property that, there is an integer constant c such
that λ(x) + λ(xy) + λ(y) = c for any xy ∈ E(G). If λ(V (G)) =
{1, 2, · · · , |V (G|} then edge-magic total labeling is called super edgemagic total labeling. In this paper we investigate super edge-magic
total labeling on w-trees.
Keywords : Super edge-magic total labeling, w-tree.
1
Introduction
All graphs in this paper are finite, simple, planar and undirected. Graph G
has the vertex-set V (G) and edge-set E(G). A general reference for graphtheoretic ideas can be seen in [15].
A labeling (or valuation) of a graph is a map that carries graph elements
to numbers (usually to positive or non-negative integers). In this paper the
domain will usually be the set of all vertices and edges and such labelings are
called total labelings. Some labelings use the vertex-set only, or the edge-set
only, and we shall call them vertex-labelings and edge-labelings respectively.
Other domains are possible. The most complete recent survey of graph
labelings can be seen in [10]. There are many types of graph labelings, for
example harmonius, cordial, graceful and antimagic.
In this paper, we focus on one type of labeling called edge-magic total
labeling. An edge-magic total labeling of a graph G is a one-to-one map
λ from V (G) ∪ E(G) onto the integers {1, 2, · · · , |V (G) ∪ E(G)|} with the
property that, there is an integer constant c such that λ(x)+λ(xy)+λ(y) =
c for any xy ∈ E(G). An edge-magic total labeling λ of a graph G is called
a super edge-magic total labeling if λ(V (G)) = {1, 2, · · · , |V (G|}.
The subject of edge-magic total labelings of graphs has its origin in the
work of Kotzig and Rosa [12, 13], on what they called magic valuations
2
K. Ali, M. Hussain, A. Razzaq
of graphs. A more restrictive form of edge-magic labeling was defined by
Enomoto et al. in [7] as super edge-magic total labelings which is an equivalent concept of strongly indexable labelings defined before by Archarya and
Hedge [3]. A number of classification studies on edge-magic total graphs has
been intensively investigated. A part of these studies results includes
–
–
–
–
–
–
Every cycle Cn is super edge-magic total if and only if n is odd [7].
Km,n is super edge-magic total if and only if m = 1 or n = 1 [7].
Kn is super edge-magic total if and only if n = 1, 2, or 3 [7].
nK2 is super edge-magic total if and only if n is odd [6].
2Pn is super edge-magic total if and only if n is not 2 or 3 [9].
The friendship graph consisting of n triangles is super edge-magic total
if and only if n is 3, 4, 5 or 7 [14].
– nP3 is super edge-magic total for even n ≥ 4 [5], and nP3 is super
edge-magic total for n odd [9].
– The fan Fn is super edge-magic total if and only if 1 ≤ n ≤ 6 [8].
– If G is a (super) edge-magic bipartite or tripartite graph, and n is odd,
then nG is (super) edge-magic [9].
Probably the most famous conjecture in the world of super edge-magic
total labeling claims that every tree is super edge-magic total [6]. To our
knowledge this conjecture remains open.
Definition 1. Let G be a graph with vertex and edge sets as follows:
V (G) = {(c1 , c2 , b, w, d) ∪ (x1 , x2 , ..., xn ) ∪ (y 1 , y 2 , ..., y n )}
E(G) = {c1 x1 , c1 x2 , ..., c1 xn }∪{c2 y 1 , c2 y 2 , ..., c2 y n }∪{c1 b, c1 w}∪{c2 w, c2 d},
we shall call it w-graph and it shall be denote by Wn .
Definition 2. Suppose that we have k copies of w-graphs Wn1 , Wn2 ,
..., Wnk , n1 ≥ n2 ≥ ... ≥ nk . A w-tree W T (n1 , n2 , n3 , ..., nk ; k) is a tree
obtained by taking a new vertex a and joining it with {dni : 1 ≤ i ≤ k},
where dni is the d vertex of w-graph Wni .
In this paper we present super edge-magic total labelings for w-trees
as well as for disjoint union of isomorphic and non isomorphic copies of
w-trees.
2
Main Results
Before introducing our main theorems, let us consider the following lemma
found in [8] that gives a necessary and sufficient condition for a graph to
be super edge-magic total.
Super edge-magic total labeling of a tree
3
Lemma 1 A graph G with v vertices and e edges is super edge-magic total
if and only if there exists a bijective function f : V (G) → {1, 2, · · · , v}
such that the set S = {f (x) + f (y)|xy ∈ E(G)} consists of e consecutive
integers. In such a case, f extends to a super edge-magic total labeling of
G with magic constant c = v + e + s, where s = min(S).
In the following sections we present super edge-magic total labelings of
w-trees and forests consisting of disjoint unions of w-trees.
2.1
Super edge-magic total labelings of w-trees
Theorem 1 Every w-tree G ∼
= W T (n1 , n2 , n3 , ..., nk ; k) admits a super
edge-magic total labeling if ni ≥ k − 2 when k is even and ni ≥ k − 1 when
k is odd, n1 ≥ n2 ≥ ... ≥ nk .
Proof.
If k = 2 then the resulting tree is a caterpillar, and every caterpillar is
super edge-magic [12]. Therefore we assume that k ≥ 3. If v = |V (G)|,
Pk
e = |E(G)| then v = 5k + 1 + 2 p=1 np and e = v − 1. We denote the
vertex and edge sets of G as follows:
V (G) = {a} ∪ {bi : 1 ≤ i ≤ k}
∪ {wi : 1 ≤ i ≤ k} ∪ {di : 1 ≤ i ≤ k}
∪ {cij : 1 ≤ i ≤ k, j = 1, 2}
∪ {xli , yil : 1 ≤ i ≤ k, 1 ≤ l ≤ ni }
E(G) = {bi ci1 : 1 ≤ i ≤ k} ∪ {wi ci1 : 1 ≤ i ≤ k} ∪ {wi ci2 : 1 ≤ i ≤ k}
∪ {di ci2 : 1 ≤ i ≤ k} ∪ {adi : 1 ≤ i ≤ k}
∪ {ci1 xli , ci2 yil : 1 ≤ i ≤ k, 1 ≤ l ≤ ni .}
Before formulating the labeling, let s =
i
P
and αi = 2
np ∀ i ∈ N .
k
2
for even k, s =
k−1
2
for odd k
p=1
Now, we define labeling λ : V → {1, 2, ..., v} as follows:
k
λ(a) = v − 2d e.
2

 λ(a) + 2i − 2(s + 1),
λ(ci1 ) =

λ(c11 ) + 2i − 2(s − 2),
f or 1 ≤ i ≤ s,
f or 1 + s ≤ i ≤ k,
4
K. Ali, M. Hussain, A. Razzaq

 λ(ci1 ) + 1,
λ(ci2 ) =

λ(ci1 ) − 1,

 αi−1 + 3i − 2,
λ(bi ) =

αi + 3i,

 λ(bi ) + ni + 1,
λ(wi ) =

λ(bi ) − ni − 1,
f or 1 ≤ i ≤ s,
f or 1 + s ≤ i ≤ k,
f or 1 ≤ i ≤ s,
f or 1 + s ≤ i ≤ k,
f or 1 ≤ i ≤ s,
f or 1 + s ≤ i ≤ k,

 λ(wi ) + ni + 2i − 2s + 1,
λ(di ) =

λ(wi ) − ni + 2i − 2s − 3,
f or 1 ≤ i ≤ s,
f or 1 + s ≤ i ≤ k,
For 1 ≤ i ≤ s,
λ(xli ) = λ(bi ) + l, f or 1 ≤ l ≤ ni
λ(yil ) = {λ(wi ) + l}\{λ(di )}, f or 1 ≤ l ≤ ni + 1.
For 1 + s ≤ i ≤ k
λ(xli ) = {λ(bi ) − l}, f or 1 ≤ l ≤ ni
λ(yil ) = {λ(wi ) − l}\{λ(di )}, f or 1 ≤ l ≤ ni
The set of all edge-sums generated by the above formula forms a consecutive integer sequence v − 2t − 2s + 1, v − 2t − 2s + 2, v − 2t − 2s +
3, ..., v + e − 2t − 2s, where t = d k2 e. Therefore, by Lemma 1 λ can be
extended to a super edge-magic total labeling and we obtain the magic
constant c = v + e + s = v + e + v − 2t − 2s + 1 = 3v − 2t − 2s.
t
u
Using theorem 2.1 found in [9], we obtain the following immediate corollary
of theorem 1:
Corollary 1 Consider a w-tree G ∼
= W T (n1 , n2 , n3 , ..., nk ; k) such that
ni ≥ k − 2 when k is even and ni ≥ k − 1 when k is odd, where k ≥ 3 and
n1 ≥ n2 ≥ ... ≥ nk . If m is an odd integer, then mG is super edge-magic.
Super edge-magic total labeling of a tree
2.2
5
Super edge-magic total labelings on m isomorphic copies of
w-trees
Theorem 2 For k ≥ 3 and m ≥ 2, G ∼
= mW T (n1 , n2 , n3 , ..., nk ; k) admits
a super edge-magic total labeling if ni ≥ 2mk − 2, 1 ≤ i ≤ k, and n1 ≥
n2 ≥ n3 ≥ ... ≥ nk .
Proof.
If v = |V (G)| and e = |E(G)| then v = 5mk +m+2m
k
P
p=1
np and e = v −m.
We denote the vertex and edge sets of G as follows:
V (G) = {ar : 1 ≤ r ≤ m} ∪ {bri : 1 ≤ i ≤ k, 1 ≤ r ≤ m}
∪ {wir : 1 ≤ i ≤ k, 1 ≤ r ≤ m}
∪ {dri : 1 ≤ i ≤ k, 1 ≤ r ≤ m}
∪ {crij : 1 ≤ i ≤ k, 1 ≤ r ≤ m, j = 1, 2}
lr
∪ {xlr
i , yi : 1 ≤ i ≤ k, 1 ≤ m ≤ h, 1 ≤ l ≤ ni }
r r r r r
E(G) = {a di , ci2 di , ci2 wir , cri1 wir , cri1 bri : 1 ≤ i ≤ k, 1 ≤ r ≤ m}
r lr
∪ {cri1 xlr
i , ci2 yi : 1 ≤ i ≤ k, 1 ≤ r ≤ m, 2, 1 ≤ l ≤ ni }
Now, we define labeling λ : V → {1, 2, ..., v} as follows:
λ(ar ) = v + r − m,
1≤r≤m
Throughout the following labeling, we consider:
i
P
1 ≤ i ≤ k, 1 ≤ r ≤ m, j = 1, 2 and αi = 2
np ∀ i ∈ N .
p=1
λ(crij )
1
= λ(a ) − 2mk + (j − 1) + 2(i − 1) + 2k(r − 1),
λ(bri ) = 3i − 2 + (r − 1)(αk + 3k) + αi−1 ,
λ(wir ) = λ(bri ) + ni + 1,
λ(dri ) = λ(wir ) + λ(cri2 ) + ni + 2 − λ(ar ),
λ(xlr
i ) = λ(bi ) + l, f or 1 ≤ l ≤ ni
λ(yilr ) = {λ(wi ) + l}\{λ(di )}, f or 1 ≤ l ≤ ni + 1.
The set of all edge-sums generated by the above formula forms the consecutive integer sequence m(3k +αk )+2, m(3k +αk )+3, · · · , m(3k +αk )+e+1.
Therefore, by Lemma 1 λ can be extended to a super edge-magic total labeling and we obtain the magic constant c = v + e + s = 3v − 2mk − 2m + 2.
t
u
For example Figure 1 shows a super edge-magic total labeling of
2W T (11, 10, 10; 3).
6
K. Ali, M. Hussain, A. Razzaq
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 65 66 67 68 69 70 71 64
147
148
26 27 28 29 30 31 32 33 34 35 36 37 38 39 41 42 43 44 45 46 47 48 40
145
146
1
2
3 4 5 6 7
8 9 10 11 12 13 24 23 14 16 17 18 19 20 21 22 25 15
143
144
155
140
122
120121 123124125126127128 129130
131132133134 135136137138139 142
141
153
154
97 98 99100101102 103104105106107108109110111112113114115117118119 116
151
152
72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 92 93 94 95
96 91
149
150
156
Fig. 1. Super edge-magic total labeling of 2W T (11, 10, 10; 3) with c = 455.
Super edge-magic total labeling of a tree
2.3
Super edge-magic total labelings of m non-isomorphic
copies of w-trees
Theorem
Sm 3 Forq m q≥ 2,q kq ≥qkq+1 ≥ 3, and 1 ≤ q ≤ m,
G∼
= q=1 W T (n1 , n2 , n3 , ..., nkq ; kq ) admits a super edge-magic
total labeling, where nqi ≥ 2mkq − 2 for 1 ≤ i ≤ kq .
Proof.
If v = |V (G)| and e = |E(G)| then,
Pm
Pm Pkq q
v = 5 q=1 kq + 2 q=1 ( p=1
np ) + m and e = v − m.
We denote the vertex and edge sets of G as follows:
V (G) = {ar : 1 ≤ r ≤ m} ∪ {bri : 1 ≤ i ≤ kr , 1 ≤ r ≤ m}
∪ {wir : 1 ≤ i ≤ kr , 1 ≤ r ≤ m}
∪ {dri : 1 ≤ i ≤ kr , 1 ≤ r ≤ m}
∪ {crij : 1 ≤ i ≤ kr , 1 ≤ r ≤ m, j = 1, 2}
rl
∪ {xrl
i , yi : 1 ≤ i ≤ kr , 1 ≤ r ≤ m, 1 ≤ l ≤ ni }
E(G) = {bri cri1 : 1 ≤ i ≤ kr , 1 ≤ r ≤ m}
∪ {wir cri1 : 1 ≤ i ≤ kr , 1 ≤ r ≤ m}
∪ {wir cri2 : 1 ≤ i ≤ kr , 1 ≤ r ≤ m}
∪ {dri cri2 : 1 ≤ i ≤ kr , 1 ≤ r ≤ m}
∪ {ar dri : 1 ≤ i ≤ kr , 1 ≤ r ≤ m}
r rl
∪ {cri1 xrl
i , ci2 yi : 1 ≤ i ≤ kr , 1 ≤ r ≤ m, 1 ≤ l ≤ ni }
Now, we define labeling λ : V → {1, 2, ..., v} as follows:
λ(ar ) = v + r − m,
1≤r≤m
For 1 ≤ i ≤ kr , 1 ≤ r ≤ m, 1 ≤ j ≤ 2
λ(crij ) = λ(a1 ) − 2
m
X
kp + (j − 1) + 2(i − 1) + 2
p=1
r
+2
λ(bri ) = 3i − 2 + βi−1
Pm
Pi−1
q=1 (
p=1
kp ,
p=1
kq
r−1 X
r−1
X
X
(
nqp ) + 3
kq ,
q=1 p=1
r
where βi−1
=
r−1
X
1 ≤ i ≤ kr ,
q=1
nqp ).
λ(wir ) = λ(bri ) + nri + 1,
1 ≤ i ≤ kr , 1 ≤ r ≤ m
λ(dri ) = λ(wir ) + λ(cri 2) + nri + 2 − λ(ar ),
1 ≤ i ≤ kr , 1 ≤ r ≤ m
r
λ(xrl
i ) = λ(bi ) + l, f or 1 ≤ l ≤ ni , 1 ≤ i ≤ kr , 1 ≤ r ≤ m
7
8
K. Ali, M. Hussain, A. Razzaq
λ(yirl ) = {λ(wi ) + l}\{λ(di )}, f or 1 ≤ l ≤ ni , 1 ≤ i ≤ kr , 1 ≤ r ≤ m.
The set of all edge-sums generated
Pm by the above formula
Pm forms the
consecutive integer sequence v − 2 p=1 kp − m + 2, v − 2 p=1 kp − m +
Pm
Pm
3, v − 2 p=1 kp − m + 4, · · · , 2v − 2 p=1 kp − 2m + 1. Therefore, by Lemma
1, λ can be extended to a super edge-magic
Pmtotal labeling and we obtain
the magic constant c = v + e + s = 3v − 2 p=1 kp − 2m + 2.
t
u
In Figure 2 and 3 we show W T (n11 , n12 , n13 ; 3) ∪ W T (n21 , n22 , n23 ; 3), and a
super edge-magic total labeling of W T (10, 11, 12; 3) ∪ W T (11, 10, 11; 3) respectively.
b
1
1
11
x
1
x
1
1n
1
x 1 w
1
1
12
1
11
y
1
1
c
11
12 13
y
y
1
1
1
1n 1
y 1 d
1
1
1
b
2
11
x
2
12
x
2
1
c
12
13
x
2
1
1n
1
x 2 w
2
2
2
1
21
x
1
2
2n
x 1
1
22
x
1
2
c
11
2
w
1
21
y
1
22
y
1
23
y
1
2
2n
y 1
1
d
2
1
b
2
2
21
x
2
22
x
2
23
x
2
2
c
12
c
13
y
2
1
1n
y 2
2
1
d
b
2
1
3
11
x
3
12
x
3
1
1n3 1
11
y
w
x
3 3
3
12
y
3
13
y
3
1
1n
y 3
3
23
y
3
2
2n
y 3
3
1
d
3
1
c
32
1
2
2n
x 2
2
2
w
2
2
21
21 22
y
y
2
2
c
a
13
x
3
1
c
31
1
c
22
1
c
21
a
b
11 12
y
y
2
2
23
y
2
2
22
2
2n
y 2
2
2
d
2
b
2
3
21
x
3
22
x
3
23
x
3
2
2n
x 2
3
2
c
31
2
Fig. 2. A forest W T (n11 , n12 , n13 ; 3) ∪ W T (n21 , n22 , n23 ; 3).
2
w
3
21
y
3
22
y
3
2
c
32
2
d
3
Super edge-magic total labeling of a tree
50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 70 71 72 73 74 75 69
49
153
154
24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 41 42 43 44 45 46 47 48
151
4 5 6 7
8 9
10 11
149
40
152
12
1 2 3
9
23 14 15 16 17 18 19 20 21 22 13
150
161
128 130 132 134
142 144
138
136
126
146
127 129 131 133 135
124 125
139 140 141 143 145
137
148 147
159
104 106 108 110 112 114
121
101102 103 105 107 109 111 113 115116117118119 122 123
160
120
158
157
76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 96 97 98 99 100 95
155
156
162
Fig. 3. Super edge-magic total
W T (11, 10, 11; 3) with c = 473.
labeling
of
forest
W T (10, 11, 12; 3) ∪
10
K. Ali, M. Hussain, A. Razzaq
References
1. K. Ali, A. Ahmad and E. T. Baskoro, On super edge-magic total labeling of
a forest of banana trees, Utilitas Math., to appear.
2. B. D. Acharya and S. M. Hedge, Arithmetic graphs, J. Graph Th., 14 (1990),
275-299.
3. B. D. Acharya and S. M. Hedge, Strongly indexable graphs, Discrete Math.,
93 (1991), 123-129.
4. A. Ahmad, A. Q. Baig and M. Imran, On super edge-magicness of graphs,
Utilitas Math., to appear.
5. E. T. Baskoro and A. A. G. Ngurah, On super edge-magic-total labeling,
Bull. Inst. Combin. Appl., 37 (2003), 82-87.
6. Z. Chen, On super edge-magic graphs, J. Combin. Math. Combin. Comput.,
38 (2001), 55-64.
7. H. Enomoto, A. S. Llado, T. Nakamigawa and G. Ringle, Super edge-magic
graphs, SUT J. Math., 34 (1980), 105-109.
8. R. M. Figueroa-Centeno, R. Ichishima and F. A. Muntaner-Batle, The place
of super edge-magic labeling among other classes of labeling, Discrete Math.,
231 (2001), 153-168.
9. R. M. Figueroa-Centeno, R. Ichishima and F. A. Mantaner-Batle, On edgemagic labeling of certain disjoint union graphs, Australas J. Combin., 32
(2005), 225-242.
10. J. A. Gallian, A dynamic survey of graph labeling, Electronic J. Combin.,
(2009).
11. M. Hussain, E. T. Baskoro and Slamin, On super edge-magic total labeling
of banana trees, Utilitas Math., 79 (2009), 243-251.
12. A. Kotzig and A. Rosa, Magic valuations of finite graphs, Canad. Math. Bull.,
13(1970), 451-461.
13. A. Kotzig and A. Rosa, Magic valuation of complete graphs, Centre de
Recherches Mathematiques, Universite de Montreal, (1972), CRM-175.
14. Slamin, M. Baca, Y. Lin, M. Miller and R. Simanjuntak, Edge-magic total
labeling of wheels, fans and friendship graphs, Bull. Inst. Combin. Appl.,
35(2002), 89-98.
15. D. B. West, An Introduction to Graph Theory (Prentice-Hall, 1996 ).