SUPER MAGIC CORONATIONS OF GRAPHS

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 10, October (2014), pp. 191-200 © IAEME
191
SUPER MAGIC CORONATIONS OF GRAPHS
Dayanand.G.K1
, Shabbir Ahmed2
, Danappa Akka3
1
Department of Mathematics BKIT (Rural Engg. College) Bhalki, Dist. Bidar Karnataka, INDIA
2
Professor (Rtd.) Department of Mathematics, Gulbarga, University Gulbrga Karnataka. India
3
Department of Mathematics, Raja Rajeshwari College of Engineering Bangalore-74 Karnataka,
INDIA
ABSTRACT
A labeling of graph G is a mapping that carries a set of graph elements into a set of numbers
(Usually positive integers) called labels. An edge magic labeling on a graph with p vertices and q
edges will be defined as a one-to-one map taking the vertices and edges onto the integers 1,2,----,
p+q with the property that the sum of the label on an edge and the labels of its end vertices is
constant independent of the choice of edge. Moreover G is said to be super magic if fሾVሺGሻሿ = {q +
1, q + 2 … … , p + q} and fሾEሺGሻሿ = {1,2, … … , q}.In this paper we show that the n-crown is super
magic for any positive integer n and thus extended what was known about the harmoniousness,
sequentialness and felicitousness of such graphs. Lastly we establish two results on CmÍKഥn for some
integer m of certain super magic graphs to obtain more super magic graphs.
Keywords: Magic, Super Magic, Edge Magic, Super Edge Magic.
1. INTRODUCTION
Let G be a finite simple undirected graph. The set of vertices and edges of a graph G will be
denoted by V(G) and E(G) respectively, p = |VሺGሻ| and q = |EሺGሻ|. For simplicity we denote V(G)
by V and E(G) by E.
A labeling (or valuation) of a graph is a map that carries graph elements to numbers (usually
to the positive or non-negative integers). In this paper, the domain will usually be the set of all
vertices and edges such a labeling are called total labeling. Some labeling use the vertex set alone or
the edge set alone and we shall call them vertex labeling and edge labeling respectively. Other
domains are possible. The most complete recent survey of graph labeling is [7].
Kotzig and Rosa [12] defined a magic labeling to be a total labeling in which the labels are
the integers from 1 to p+q. The sum of labels on an edge and its two end vertices is constant. In 1996
Ringel and Llado [16] redefined this type of labeling (and called the labeling; edge magic, causing
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some confusion with papers that have followed the terminology of [13] mentioned below) see also
[8,9]. Recently Enomoto et.al [6] have introduced the same super edge magic for magic labeling in
the sense of Kotzig and Rosa with the added property that the p vertices receive the smaller labels
{1,2,------,p}. In 2010, Akka and Nanda [1] introduced the same type super magic for magic labeling
with added property that the p vertices receive the bigger labels {q+1,q+2,-----,q+p} and q edges
receive the smaller labels {1,2,----,q}
An edge magic total (EMT) labeling is a one to one mapping f from VUE onto the integers
1,2,----, p+q with the property that for every (x, y) in E.
f(x) + f(y) + f(xy)=k for some constant k. A graph that has an edge magic total labeling is called an
edge magic total graph. An edge magic total labeling is called a super edge magic total (SEMT)
labeling [6] if f(V)= {1,2,---p} and edge magic total labeling is called a super magic total labeling
(SMT) [1] if f(V)={q+1,q+2,….., q+p} and a graph that has SEMT (SMT) labeling is called a SEMT
(SMT) graph. Research in SEMT (SMT) labeling has been particularly popular during the last
decade. For details see the Gallian’s dynamic survey [7].
In 1983, Lih [14] introduced magic labeling of planar graphs where labels extended to faces
as well as edges and vertices an idea which he traced back to 13th
century Chinese roots. Baca (see
the example [3,4]) has written extensively on these labelings. A some what related sort of magic
labeling was defined by Dickson and Rogers in [5].
Lee, Seah and Tan [13] introduced a weaker concept which they called edge-magic in 1992.
The edges are labelled and the sums of the vertices are required to be congruent modulo the number
of vertices.
Total labeling have also studied in which the sum of the labels of all edges adjacent to the
vertex x plus the label of x itself is constant. A paper on these labelings is in preparation [15].
In order to clarify the terminological confusion defined above, we define a labeling to be
edge-magic if the sum of all labels associated with an edge equals a constant independent of the
choice of edge and vertex magic if the same property holds for vertices (This terminology could be
extended to other sub structures; face- magic for example). The domain of the labeling is specified
by a modifier on the word “labeling”. We shall always require that the labeling is a one-to-one map
onto the appropriate set of consecutive integers from1. For example, Stewart studies vertex magic
edge-labelings and Kotzig and Rosa define edge magic total labelings. Hypermagic labelings are
vertex-magic total labelings.
The following result found in [2] provides a necessary and sufficient condition for graph to
be super magic
Lemma 1.1 A (p , q) graph G is super magic if and only if there exists a bijective function f: VሺGሻ →
{q + 1, q + 2, … … . . , +q + p}such that the set
S = {fሺxሻ + fሺyሻ/xy ∈ EሺGሻ} consists of q consecutive integers. In such a case, f extends to a super
magic labeling of G with valence kl
= q+s where s=min and
S = {fሺxሻ + fሺyሻ/xy ∈ EሺGሻ}
= {k୪
− 1, k୪
− 2, … … . , k୪
− q}
Lemma 1.2 [2] If a (p , q) graph G is Super magic then G is felicitous whenever it is a tree or
satisfies q ≥ p
Thus we establish the following relationship between super magic labelings and felicitous
labelings since every harmonious labeling is certainly a felicitous labeling.
Lemma 1.3 [2] If a (p, q) graph G is super magic then G is felicitous whenever it is tree or satisfies
q≥p.
Through out this paper we will utilize the following definitions.

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The Corona product G1ÍG2 of two graphs G1 and G2 defined as the graph obtained by taking
one copy of G1 (which has order p1) and p1 copies of G2 and then joining the i-th vertex of G1 to
every vertex in the i-th copy of G2. The graph CmÍKഥn is called the n-crown with cycle length m. We
will refer to the 1-crown as the crown with cycle length m. For the sake of previty, we will refer to
the n-crown with cycle length m simply as the n-crown if its cycle length is clear from the
context.
A (p, q) graph G with q ≥ p is harmonious [11] if there exists an injective function f: VሺGሻ →
Z୯ such that each edge xy of G is labeled fሺxሻ + fሺyሻሺmod q) and the resulting labels are distinct.
Such a function is called a harmonious labeling. If G is a tree (so that q = p - 1) exactly two vertices
are labeled the same, otherwise the definition is the same. For a (p, q) graph G, an injective function
f: V(G) → {0,1, … . . , q − 1} is a sequential labeling of G if each edge xy of G is labeled f(x) + f(y)
and the resulting edge labels are {m, m+1, ----, m+q-1} for some positive integer m. If such a
labeling exists, then G is said to be sequential [10]. In case of a tree, Grace permits the vertex labels
to range from o to q with no vertex label used twice. A graph G of size q is felicitous if there exists
an injective function f: V(G) → Z୯ାଵ such that each edge xy of G is labeled f(x) + f(y)(mod q) and
the resulting edge labels are distinct. Such a function is called a felicitous labeling [17].
In this paper, we show that a cycle (of length m) with ‘n’ pendent edges attached at each
vertex, known as an n-crown CmÍKഥn is super magic, harmonious, sequential and felicitous. Also we
establish two results that illustrate how attaching pendant edges to the vertices of certain super magic
graphs will result in more super magic graphs.
2. CORONATION OF SUPER MAGIC GRAPHS
Theorem 2.1 Let G be a graph of odd order p ≥ 3 for which ther exists a super magic labeling f with
the property that,
max{f(u) + f(v)/uv ∈ E(G)} = p − 1
Then GÍKഥn is super magic for every positive integer n.
Proof Let f be a super magic labeling of G with valence k' and supposing that f has the property that
f(v୧) = q + i for every integer i with 1 ≤ i ≤ p where
V(G) = {v୧/1 ≤ i ≤ p} and E(G) = q
Further let V(GÍKഥn) = V(G) ∪ {w୧
୨
/1 ≤ i ≤ p and 1 ≤ j ≤ n}
and E(GÍKഥn) = E(G) ∪ {v୧w୧
୨
/1 ≤ i ≤ p and 1 ≤ j ≤ n}.
Then, define the vertex labeling
g: V(GÍKഥn) → {q + 1, q + 2, … … q + p(n + 1)} such that g(v) = f(v) for every vertex v of G and
g൫w୧
୨
൯ =
2pn + q + 1 − i −
୮(ଶ୨ିଵ)ାଵ
ଶ
if 1 ≤ i ≤
୮ିଵ
ଶ
, 1 ≤ j ≤ n
(2n + 1)p + q + 1 − i −
ଶ
if
୮ାଵ
ଶ
≤ i ≤ p, 1 ≤ j ≤ n
To prove that g extends to a super magic labeling of GÍKഥn, consider the set
S୧
୨
= {f(v୧) + f(w୧
୨
)/ 1 ≤ i ≤ p and 1 ≤ j ≤ n}

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and let m୨ = min൛S୧
୨
/1 ≤ i ≤ pൟ = 2(np + p + q) + p + 1 +
ଶ

M୨ = Min{ S୧
୨
/1 ≤ i ≤ p} = 2(np + p + q) + 2p +
p(2j − 1) + 1
2
Finally observe that mଵ = 2(np + p + q) +
ଷ୮ାଵ
ଶ
+ 1, M୨ + 1 = m୨ାଵ
(i ≤ j ≤ n − 1) and S୧
୨
is a set of consecutive integer 1 ≤ i ≤ p and 1 ≤ j ≤ n) which implies that g
extends to a super magic labeling of GÍKഥn with valence k'+4np
By lemmas 1.2 and 1.3 we obtained the following corollary:
Corollary 2.2 Let G be a graph of odd order p ≥ 3 for which there exists a super magic labeling f
with the condition that
max {f(u) + f(v) ∈ E(G)} = p + q + 1 −
ଷ୮ାଵ
ଶ
Then GÍKഥn is harmonious sequential and felicitous for every positive integer n.
We now provide a similar, though in a sense weaker result for graphs with even order at least 4
Theorem 2.3 Let G be a graph of even order p ≥ 4 having a super magic labeling f with the
condition that
max {f(u) + f(v)/uv ∈ E(G)} = p + q + 1 −
ଷ୮
ଶ
Then the graph H got by attaching n edges to each vertex of G except the vertex v with f(v) = q + p
super magic for every positive integer n
Proof Let V(G) = {vଵ, vଶ, − − −, v୮} then super magic labeling of G with valence kଵ
satisfying the
condition that f(v୧) = q + i for i = 1,2, − − −, p. Now define the graph H as follows:
V(H) = V(G) ∪ {w୧
୨
/1 ≤ i ≤ p − 1 and 1 ≤ j ≤ n}
and E(H) = E(G) ∪ {v୧w୧
୨
/1 ≤ i ≤ p − 1 and 1 ≤ j ≤ n}
One can use an analogous argument to the one used in the proof of the previous theorem the
vertex labeling
g: V(H) = {q + 1, q + 2, −−, q + p(n + 1) − n} such that g(v) = f(v) for every vertex v of G and
g൫w୧
୨
൯ =
2np + q + 1 − i −
ଶ
if 1 ≤
୮ିଵ
ଶ
, 1 ≤ j ≤ n
(2n + 1)p + q + 1 − i −
ଶ
if
୮
ଶ
≤ i ≤ p − 1, 1 ≤ j ≤ n
extends to a super magic labeling of H with valence kଵ
+ 4n(p − 1)
Again Lemmas 1.2 and 1.3 we have the following corollary

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Corollary2.4 Let G be a graph of even order p ≥ 4 having a super magic labeling f with condition
that
max{f(u) + f(v)/uv ∈ E(G)} = p + q + 1 −
3p
2
Then the graph H obtained by attaching n pendent edges to each vertex of G except the vertex v with
f(v) = p + q is harmonious, sequential and felicitous for every positive integer n.
3. SUPER MAGIC LABELINGS OF N-CROWNS
To proceed to study the super magicness of n-crowns we start with a result pertaining 2-
regular graphs.
Theorem 3.1 If G is a super magic 2-regular, then GÍKഥn is super magic for every positive integer n.
Proof Let f be a super magic labeling of G with valence k. Suppose that H is a component of GÍKഥn.
Then H ≅ C୰ÍKഥn for some integer r ≥ 3.
Let V(H) = {v୧/i ∈ Z୰} ∪ {u୧,୨/i ∈ Z୰ and 1 ≤ j ≤ n}
and E(H) = {v୧v୧ାଵ/i ∈ Z୰} ∪ {v୧u୧,୨/i ∈ Z୰ and 1 ≤ j ≤ n}
where Z୰ denotes the set of integers modulo r.Then f/H extends to a labeling g of H as follows:
g(v୧) = (n + 1)ሾ2r − f(v୧) + 1ሿ,
g(v୧ିଵv୧) = (n + 1)ሾ2r − f(v୧ିଵv୧)ሿ
g൫u୧,୨൯ = (n + 1)ሾ2r − f(v୧ିଵ) + 1ሿ − j,
g൫v୧u୧,୨൯ = (n + 1)ሾ2r − f(v୧ିଵv୧)ሿ + 1 + j
when i ∈ Z୰ and 1 ≤ j ≤ n. Therefore f extends likewise in every component of GÍKഥn and a super
magic labeling of GÍKഥn is obtained with valence (n + 1)(6r − k + 2) + 1.
Theorem 3.2 [1] The n-cycle c୬ is super magic if and only if n ≥ 3 is odd.
In the following result, we show with considerable more effort that the n-crowns with cycle
of length m are super magic when m ≥ 4 is even.
Theorem 3.3 For every two integers m ≥ 3 and n ≥ 1, the n-crown
G ≅ C୫ÍKഥn is super magic
Proof Let G ≅ C୫ÍKഥn be the n-crown with
V(G) = {u୧/1 ≤ i ≤ m} ∪ {v୧୨/1 ≤ i ≤ m and 1 ≤ i ≤ n}
and E(G) = {uଵu୫} ∪ {u୧u୧ାଵ/1 ≤ i ≤ m − 1} ∪ {u୧v୧,୨/1 ≤ i ≤ m and 1 ≤ i ≤ n}

196
Now, notice that if m ≥ 3 is odd then the result follows from theorem 3.1 and 3.2. Thus assume that
m ≥ 4 is even for the reminder of the proof and proceed by cases by means of Lemma 1.1
Case I For m=4, define the vertex labeling
f: V(G) → {4(n + 1) + 1, … … . ,8(n + 1)} such that
f(uଶ୧ିଵ) = 2m(n + 1) + 1 − i f(uଶ୧) = 2m(n + 1) + 1 − 3i
f൫vଶ୧ିଵ,ଵ൯ = 2m(n + 1) − 2 − 2i f൫vଶ୧,ଵ൯ = 2m(n + 1) − 11 + 4i
When i = 1 or 2 and f൫v୧,୨൯ = 2m(n + 1) − 4 − 4j + i 1 ≤ i ≤ 4, 2 ≤ j ≤ n
Case II For m=6, define the vertex labeling
f: V(G) → {6(n + 1) + 1,6(n + 1) + 2, … … . , +12(n + 1)} Such that
f(uଵ) = 12(n + 1) − 8 f(uଶ) = 12(n + 1) f(uଷ) = 12(n + 1) − 3
f(uସ) = 12(n + 1) − 1 f(uହ) = 12(n + 1) − 4 f(u଺) = 12(n + 1) − 2
f൫vଵ,ଵ൯ = 12(n + 1) − 5 f൫vଶ,ଵ൯ = 12(n + 1) − 7 f൫vଷ,ଵ൯ = 12(n + 1) − 6
f൫vସ,ଵ൯ = 12(n + 1) − 12 f൫vହ,ଵ൯ = 12(n + 1) − 10 f൫v଺,ଵ൯ = 12(n + 1) − 9
and f൫v୧,୨൯ =
12(n + 1) + 5(i − 1) − 6j if 1 ≤ i ≤ 2 and 2 ≤ j ≤ n
12(n + 1) + 2 − i − 6j if 3 ≤ i ≤ 6 and 2 ≤ j ≤ n
Case III For m=8 define the vertex labeling
f: V(G) → {8(n + 1) + 1, 8(n + 1) + 2, … … + 16(n + 1)}Such that
f(uଵ) = 12(n + 1) f(uଶ) = 12(n + 1) − 4 f(uଷ) = 12(n + 1) − 1
f(uସ) = 12(n + 1) − 5 f(uହ) = 12(n + 1) − 2 f(u଺) = 12(n + 1) − 6
f(u଻) = 12(n + 1) − 3 f(u଼) = 12(n + 1) − 11
f൫vଵ,ଵ൯ = 12(n + 1) − 10 f൫vଶ,ଵ൯ = 12(n + 1) − 12 f൫vଷ,ଵ൯ = 12(n + 1) − 14
f൫vସ,ଵ൯ = 12(n + 1) − 13 f൫vହ,ଵ൯ = 12(n + 1) − 15 f൫v଺,ଵ൯ = 12(n + 1) − 7
f൫v଻,ଵ൯ = 12(n + 1) − 9 f൫v଼,ଵ൯ = 12(n + 1) − 8
and f൫v୧,୨൯ = 12(n + 1) − 8(j + 1) + 1 if 1 ≤ i ≤ 8 and 2 ≤ j ≤ n
Case IV Let m=8k+2 where k is a positive integer and define the vertex labeling f: V(G) →
{(8k + 2)(n + 1) + 1, … … . . , +2(8k + 2)(n + 1)} such that

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f(u୪) =
2(8k + 2)(n + 1) − 2 − 12k if l = 1
2(8k + 2)(n + 1) + 1 − 4k − i if l = 2i − 1, 2 ≤ i ≤ 4k + 1
2(8k + 2)(n + 1) + 1 − i if l = 2i, 1 ≤ i ≤ 4k + 1
f൫v୪,ଵ൯ =
2(8k + 2)(n + 1) − 8k − i if l = 2i − 1, 1 ≤ i ≤ 2k + 2
2(8k + 2)(n + 1) − 1 − 12k if l = 2
2(8k + 2)(n + 1) − 1 − i − 12k if l = 2i, 2 ≤ i ≤ 2k
2(8k + 2)(n + 1) − 3 − 2i + 14k if l = 4k + 4i − 2, 1 ≤ i ≤ k
2(8k + 2)(n + 1) − 4 + i − 14k if l = 4k + i + 3, 1 ≤ i ≤ 2
2(8k + 2)(n + 1) − 1 − 2i − 10k if l = 4k + 4i + 3, 1 ≤ i ≤ k
2(8k + 2)(n + 1) − 2 − 2i − 14k if l = 4k + 4i + 4, 1 ≤ i ≤ k − 1
2(8k + 2)(n + 1) − 2 − 2i − 10k if l = 4k + 4i + 5, 1 ≤ i ≤ k − 1
2(8k + 2)(n + 1) − 2 − 16k if l = 8k + 2

and for 2 ≤ j ≤ n we have that
f൫vଶ୧ିଵ,୨൯ =
2(8k + 2)(n + 1) + 1 − i − 2(4k + 1)j if 1 ≤ i ≤ 2k + 1
2(8k + 2)(n + 1) − i − 2(4k + 1)j if 2k + 2 ≤ i ≤ 4k − 1
f൫vଶ୧,୨൯ =
2(8k + 2)(n + 1) + 1 − i − (4k + 1)(2j + 1) if 2 ≤ i ≤ 2k
2(8k + 2)(n + 1) + 2 − i − (4k + 1)(2j + 1) if 2k + 2 ≤ i ≤ 4k + 1
f൫vଶ,୨൯ = 2(8k + 2)(n + 1) + 1 − 2(4k + 1)(j + 1) and
f൫vସ୩ାଶ,୨൯ = 2(8k + 2)(n + 1) + 1 − 2(k + 1) − 2(4k + 1)j
Case V Let m=8k+4, where k is a positive integer and define the vertex labeling.
f: V(G) → {(8k + 4)(n + 1) + 1, … … . ,2(8k + 4)(n + 1)}
f(u୪) =
2(8k + 4)(n + 1) + 1 − i if l = 2i − 1, 1 ≤ i ≤ 4k + 2
2(8k + 4)(n + 1) − 1 − i − 4k if l = 2i, 1 ≤ i ≤ 4k + 1
2(8k + 4)(n + 1) − 5 − 12k if l = 8k + 4
f൫v୪,ଵ൯ =
2(8k + 4)(n + 1) − 4 − 12k if l = 1
2(8k + 4)(n + 1) − 7 + 4i − 16k if l = 4i − 2, 1 ≤ i ≤ k
2(8k + 4)(n + 1) − 8 + 4i − 16k if l = 4i − 1, 1 ≤ i ≤ k
2(8k + 4)(n + 1) − 9 + 4i − 16k if l = 4i, 1 ≤ i ≤ k
2(8k + 4)(n + 1) − 6 + 4i − 16k if l = 4i + 1 , 1 ≤ i ≤ k
2(8k + 4)(n + 1) − 3 − 8k if l = 4k + 2
2(8k + 4)(n + 1) − 7 − 16k if l = 4k + 3
2(8k + 4)(n + 1) − 6 − 16k if l = 8k + 3
2(8k + 4)(n + 1) − 4 − 8k if l = 8k + 4
2(8k + 4)(n + 1) − 4 + i − 12k if l = 4k + i + 3, 1 ≤ i ≤ 4k − 1
and f൫v୧,୨൯ = 2(8k + 4)(n + 1) + i − 4(2k + 1)(j + 1)if 1 ≤ i ≤ 8k + 4,2 ≤ j ≤ n

198
Case VI m=8k+6 where k is a positive integer and define the vertex labeling
f: V(G) → {(8k + 6)(n + 1) + 1, … … … . . +2(8k + 6)(n + 1)}
f(u୪) =
2(8k + 6)(n + 1) − 8 − 12k if l = 1
2(8k + 6)(n + 1) − 1 − i − 4k if l = 2i − 1,2 ≤ i ≤ 4k + 3
2(8k + 6)(n + 1) + 1 − i if l = 2i, 1 ≤ i ≤ 4k + 3
f൫v୪,ଵ൯ =
2(8k + 6)(n + 1) − 4 − i − 8k if l = 2i − 1,1 ≤ i ≤ 2k + 3
2(8k + 6)(n + 1) − 7 − 12k if l = 2
2(8k + 6)(n + 1) − 7 − i − 12k if l = 2i, 2 ≤ i ≤ 2k + 1
2(8k + 6)(n + 1) − 13 + 2i − 14k if l = 4k + 3i + 1,1 ≤ i ≤ 2
2(8k + 6)(n + 1) − 11 − 2i − 14k if l = 4k + 4i + 2,1 ≤ i ≤ k
2(8k + 6)(n + 1) − 8 − 2i − 14k if l = 4k + 4i + 4,1 ≤ i ≤ k
2(8k + 6)(n + 1) − 6 − 2i − 10k if l = 4k + 4i + 5,1 ≤ i ≤ k
2(8k + 6)(n + 1) − 7 − 2i − 10k if l = 4k + 4i + 7,1 ≤ i ≤ k − 1
2(8k + 6)(n + 1) − 10 − 16k if l = 8k + 6
and for2 ≤ j ≤ n, we have that
f൫v୪,୨൯ =
2(8k + 6)(n + 1) − i − 2(4k + 3)j + 1 if l = 2i − 1, 1 ≤ i ≤ 4k + 3
2(8k + 6)(n + 1) − i − (4k + 3)(2j + 1) + 1 if l = 2i, 1 ≤ i ≤ 4k + 3
Case VII Let m=16k, where k is a positive integer and defined the vertex labeling
f: V(G) → {16k(n + 1) + 1, … … … … ,32k(n + 1)} Such that,
f(u୪) =
32k(n + 1) + 1 − i if l = 2i − 1 and 1 ≤ i ≤ 8k
8k(4n + 1) + 1 − i if l = 2i and 1 ≤ i ≤ 8k − 1
8k(4n + 1) + 1 if l = 16k
f൫vଵ,ଵ൯ = 8k(4n + 1) − 10 f൫vଶ,ଵ൯ = 16k(2n + 1) − 10 f൫vଷ,ଵ൯ = 32kn − 14
f൫v୪,୪൯ =
32kn + 2i if l = 2i − 1 and 3 ≤ i ≤ 4k
32kn − 1 + 2i if l = 2i and 2 ≤ i ≤ 4k
32kn − 2 + 3i if l = 8k + 2i − 1 and 1 ≤ i ≤ 2
8k(4n + 1) − 4 + 8i if l = 8k + 8i − 6 and 1 ≤ i ≤ k
8k(4n + 1) − 5 + 8i if l = 8k + 8i − 4 and 1 ≤ i ≤ k
8k(4n + 1) − 3 + 8i if l = 8k + 8i − 3 and 1 ≤ i ≤ k
8k(4n + 1) + 1 + 8i if l = 8k + 8i − 2 and 1 ≤ i ≤ k
8k(4n + 1) − 1 + 8i if l = 8k + 8i − 1 and 1 ≤ i ≤ k
8k(4n + 1) + 8i if l = 8k + 8i and 1 ≤ i ≤ k
8k(4n + 1) − 2 + 8i if l = 8k + 8i + 1 and 1 ≤ i ≤ k − 1
8k(4n + 1) + 2 + 8i if l = 8k + 8i + 3 and 1 ≤ i ≤ k − 1
and f൫v୧,୨൯ = 16k(2n − j + 1)if 1 ≤ i ≤ 16k and 2 ≤ j ≤ n

199
Case VIII Let m=16k+8, where k is a positive integer and define the vertex labeling f: V(G) =
{8(2k + 1)(n + 1) + 1, … … … . ,16(2k + 1)(n + 1)}
Such that
f(u୪) =
16(2k + 1)(n + 1) + 1 − i if l = 2i − 1, 1 ≤ i ≤ 8k + 4
16(2k + 1)(n + 1) − 3 − 8k − i if l = 2i, 1 ≤ i ≤ 8k + 3
16(2k + 1)(n + 1) − 11 − 24k if l = 16k + 8
f൫v୪,ଵ൯ = 16(2k + 1)(n + 1) − 10 − 24k
f൫vଶ,ଵ൯ = 16(2k + 1)(n + 1) − 10 − 16k
f൫vଷ,ଵ൯ = 16(2k + 1)(n + 1) − 14 − 32k
f൫v୪,ଵ൯ =
16(2kn − 1) + 2i if l = 2i − 1 and 3 ≤ i ≤ 4k + 2
32kn − 17 + 2i if l = 2i and 2 ≤ i ≤ 4k + 2
2(16kn − 9) + 3i if l = 8k + 2i + 3 and 1 ≤ i ≤ 2
8k(4n + 1) − 15 + 8i if l = 8k + 8i − 2 and 1 ≤ i ≤ k + 1
8k(4n + 1) − 10 + i if l = 8k + i + 7 and 1 ≤ i ≤ 2
8k(4n + 1) − 12 + 8i if l = 8k + 8i + 2 and 1 ≤ i ≤ k
8k(4n + 1) − 14 + 8i if l = 8k + 8i + 3 and 1 ≤ i ≤ k
8k(4n + 1) − 13 + 8i if l = 8k + 8i + 4 and 1 ≤ i ≤ k
8k(4n + 1) − 11 + 8i if l = 8k + 8i + 5 and 1 ≤ i ≤ k
8k(4n + 1) − 9 + 8i if l = 8k + 8i + 7 and 1 ≤ i ≤ k
8k(4n + 1) − 8 + 8i if l = 8k + 8i + 8 and 1 ≤ i ≤ k
8k(4n + 1) − 10 + 8i if l = 8k + 8i + 9 and 1 ≤ i ≤ k − 1
and f൫v୧,୨൯ = 32k(n + 1) − (16k + 8)(j + 1) + i if 1 ≤ i ≤ 16k + 8, 2 ≤ j ≤ n
Therefore, by Lemma 1.1, f extends to a super magic labeling of G with valence 48(2k + 1)(n +
1) −
୫(ସ୬ାହ)
ଶ
+ 1
Using the relationships between super magic labelings and other labelings mentioned in the
introduction. We complete with the following corollary which settles a conjecture by
Yegnanarayanan [18]
Corollary 3.2 For every two integers m ≥ 3 and n ≥ 1, the n-crown G ≅ C୫ÍKഥnis harmonious,
sequential and felicitous.
REFERENCES
[1]. D.G. Akka and Nanda Warad, Super magic strength of a graph, Indian J. Pure and applied
Math; 41(4)(2010)557-568.
[2]. D.G. Akka, Some properties of super magic graphs (To Appear).
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[4]. M. Baca, Labeling of two classes of plane graphs, Acta Math. Appl. Sinica (English Seri.)
9(1993)82-87.
[5]. T.J. Dickson and D.G. Rogers, Problems in graph theory, V. magic valuations, South East
Asian Bull. Math.3 (1979)40-43.

200
[6]. Enomoto, Llado, Nakamigawa and Ringel, Super edge magic graphs, SUT J.Math.
2(1998)105-109.
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Discrete Math. 1(1980)382-404.
[12]. A. Kotzig and A. Rosa, Magic valuation of finite graphs, Canad. Math. Bull.13(1970)
451-461
[13]. S.M. Lee, E Seah and S.K. Tan, On edge magic graphs, Congressus Num., 86(1992)
179-191
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179-191
[15]. S.MeDougall, M. Miller, Slamin and W.D. Wallis, Vertex magic total labelings Utilitas
Math., in.
[16]. G. Ringel and A.S. Llado, Another tree conjecture, Bull. ICA 18(1996)83-85
[17]. S.C. Shee, On harmonious and related graphs, Ars Combin., 23(1987)A, 237-247
[18]. V. Yegnanarayanan, On some additive analogues of complete graphs, Center de recherches
Mathematiques Universite de Montreal (1992)CRM-175.
[19]. K. Dhanalakshmi and B. Maheswari, “Accurate and Total Accurate Dominating Sets of
Interval Graphs”, International Journal of Computer Engineering & Technology (IJCET),
Volume 5, Issue 1, 2014, pp. 85 - 93, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375.
[20]. M.Manjuri and B.Maheswari, “Strong Dominating Sets of Strong Product Graph of Cayley
Graphs with Arithmetic Graphs”, International Journal of Computer Engineering &
Technology (IJCET), Volume 4, Issue 6, 2013, pp. 136 - 144, ISSN Print: 0976 – 6367,
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