1. INTRODUCTION
All graphs in this paper are finite, simple and undirected. Terms
not defined here are used in the sense of Harary [1]. The symbols V(G)
will denote the vertex set and edge set of a graph G. The cardinality of
the vertex set is called the order of G. The cardinality of the edge set
is called the size of G. A graph with p vertices and q edges is called a
(p,q) graph.
Lo [3] introduced the notion of edge graceful graphs. A graph G
with q edges and p vertices is said to be edge graceful if there exists
a bijection f from the edge set to the set {1,2, ..., q} so that the
induced mapping [f.sup.+] from the vertex set to the set {0,1,2, ...,
p-1} given by [f.sup.+](x) = [summation]{f(xy)/xy [member of]E(G)} (mod
p) is a bijection.
The necessary condition for a graph to be edge graceful is q(q+1)
[equivalent to] 0 or p/2 (mod p). With this condition one can verify
that even cycles, and paths of even length are not edge graceful. But
whether trees of odd order are edge graceful is still open. On these
lines, we define a new type of labeling called strong edge graceful
labeling by relaxing its range through which we can get edge graceful
labeling of odd order trees for some family of graphs.
A (p,q) graph G is said to have strong edge graceful labeling if
there exists an injection f from the edge set to {1, 2, ..., [3q/2]} so
that the induced mapping [f.sup.+] from the vertex set to {0,1, ..., 2
p-1} defined by [f.sup.+](x) = [summation]{f(xy)}/xy [member of]E(G)}
(mod 2p) are distinct. A graph G is said to be strong edge graceful if
it admits a strong edge graceful labeling. In this paper, we investigate
strong edge graceful labeling (SEGL) of some graphs.
2. MAIN RESULTS
Theorem 2.1: [C.sub.n] is strong edge graceful for all n where n is
odd and n [greater than or equal to] 3.
Proof: Let [v.sub.1], [v.sub.2], ..., [v.sub.n] be the vertices of
[C.sub.n] and [e.sub.1], [e.sub.2], ..., [e.sub.n] be the edges of
[C.sub.n] denoted as in fig.1
[FIGURE 1 OMITTED]
We first label the edges [C.sub.n] of as follows
f ([e.sub.i]) = i for i = 1 to n. Then the induced vertex labels
are
[f.sup.+] ([v.sub.1]) = n+1; [f.sup.+] ([v.sub.i]) = 2i-1 for i =
1,2,3, ..., n
Clearly {[f.sup.+] ([v.sub.i]): i = 1 to n} are distinct. Hence,
[C.sub.n] is strong edge graceful for all n odd and n [greater than or
equal to] 3. The strong edge graceful labeling of [C.sub.7] and
[C.sub.11] are given below.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
Theorem 2.2: [C.sub.n] is strong edge graceful for all n where n is
even and n [greater than or equal to] 4.
Proof: Let [v.sub.1], [v.sub.2], ..., [v.sub.n] be the vertices of
[C.sub.n] and [e.sub.1], [e.sub.2], ..., [e.sub.n] be the edges of
[C.sub.n] denoted as in fig. 1. We first label the edges of [C.sub.n] as
follows
f([e.sub.i]) = i for i = 1 to n-1 ; f([e.sub.n]) = n+1. Then the
induced vertex labels are
[f.sup.+] ([v.sub.1]) n + 2; [f.sup.+] ([v.sub.n]) = 0
[f.sup.+] ([v.sub.2]) = 3; [f.sup.+] ([v.sub.i]) = [f.sup.+]
([v.sub.i-1]) + 2 for i = 3 to n-1
Clearly {[f.sup.+] ([v.sub.i]: i = 1 to no} are distinct. Hence,
[C.sub.n] is strong edge graceful for all n even and n [greater than or
equal to] 4. The strong edge graceful labeling of [C.sub.6] and
[C.sub.8] are illustrated in fig. 4 and fig.5 respectively.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
Theorem 2.3: [P.sub.n] is strong edge graceful for all n odd and n
[greater than or equal to] 3.
Proof: Let [v.sub.1], [v.sub.2], ..., [v.sub.n] be the vertices of
[P.sub.n] and {[e.sub.1], [e.sub.2], ..., [e.sub.n-1] be the edges of
[P.sub.n] as defined in the fig. 6.
[FIGURE 6 OMITTED]
We first label the edges of [P.sub.n] as f([e.sub.i]) = i for i = 1
to n-1 Then the induced vertex labels are [f.sup.+] ([v.sub.1]) = 1;
[f.sup.1] ([v.sub.1]) = 2i-1 for i = 2 to n-1.; [f.sup.+] ([v.sub.n]) =
n-1
Clearly {[f.sub.+]([v.sub.i]): i = 1 to n} are all distinct. Hence
[P.sub.n] is strong edge graceful for all n odd and n [greater than or
equal to] 3. The strong edge graceful labeling of [P.sub.9] and
[P.sub.13] are illustrated fig. 7 and fig. 8 respectively.
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
Theorem 2.4: [P.sub.n] is strong edge graceful for all n even and n
[greater than or equal to] 4.
Proof: Let [v.sub.1], [v.sub.2] ..., [v.sub.n] be the vertices of
[P.sub.n] and {[e.sub.1], [e.sub.2], ..., [e.sub.n-1]} be the edges of
[P.sub.n] as defined in the fig. 6. We first label the edges of
[P.sub.n] as follows:
f ([e.sub.i]) = i for i = 1 to n - 2; f([e.sub.n-1]) = n Then the
induced vertex labels are
[f.sup.+] ([v.sub.i]) = 2i - 1 for i = 1 to n-2.; [f.sup.+]
([v.sub.n-i]) = 2n-2; [f.sup.+] ([v.sub.n]) = n. Clearly
{[f.sup.+] ([v.sub.i]): i = 1 to n} are all distinct. Hence
[P.sub.n] is strong edge graceful for all n even and n [greater than or
equal to] 4. The strong edge graceful labeling of [P.sub.6] and
[P.sub.8] are illustrated fig. 9 and fig. 10 respectively.
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
Theorem 2.5: [C.sup.+.sub.n] is strong edge graceful for all n
[greater than or equal to] 3.
Proof: Let {[v.sub.1], [v.sub.2] ..., [v.sub.n]} [union]
{[v'.sub.1], [v'.sub.2] ..., [v'.sub.n]} be the vertices
of [C.sup.+.sub.n]. The edges of [C.sup.+.sub.n] are {[e.sub.i] =
[v.sub.i] [v.sub.i+1]: i = 1 to n - 1} [union] {[e.sub.n] = [v.sub.n]
[v.sub.1]} [union] {[v.sub.i] [v'.sub.i]: i = 1 to n} are denoted
as in fig.11
[FIGURE 11 OMITTED]
We first label the edges of [C.sup.+.sub.n] as follows:
f([e.sub.i]) = i for i = 1 to n; f([v.sub.i] [v'.sub.i]) = 2n- i+1
for i = 1 to n
Then the induced vertex labels are
[f.sup.+]([v.sub.i]) = 2n + 1 + i for i = 1 to n. ;
[f.sup.+]([v'.sub.i]) = 2n + 1 - i for i = 1 to n.
Clearly {[f.sup.+]([v.sub.i]), [f.sup.+]([v'.sub.i])} are
distinct. Hence, [C.sup.+.sub.n] is strong edge graceful for all n
[greater than or equal to] 3.
The labeling of [C.sup.+.sub.7] are [C.sup.+.sub.8] illustrated in
fig. 12 and fig. 13 respectively.
[FIGURE 12 OMITTED]
[FIGURE 13 OMITTED]
Theorem 2.6: [K.sub.1,2n] is strong edge graceful for all n
[greater than or equal to] 1.
Proof: Let {v, [v.sub.1], [v.sub.2], ..., [v.sub.2n]} be the
vertices of [K.sub.1,2n] and {[e.sub.1], [e.sub.2], [e.sub.3], ...,
[e.sub.2n]} be the edges of [K.sub.1,2n] denoted as in fig. 14.
[FIGURE 14 OMITTED]
We first label the edges of [K.sub.1,2n] as f([e.sub.i]) = i for i
= 1, 2, ..., 2n. Then the induced vertex labels are [f.sup.+]([v.sub.i])
= i for i = 1, 2, ..., 2n
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Then the induced vertex labels are all distinct. Hence,
[K.sub.1,2n] is strong edge graceful for all n [greater than or equal
to] 1. The strong edge graceful labeling of [K.sub.1,6] and [K.sub.1,8]
are illustrated in fig.15 and fig 16 respectively.
[FIGURE 15 OMITTED]
[FIGURE 16 OMITTED]
Theorem 2.7: [K.sub.1,4n-1] is strong edge graceful for all n
[greater than or equal to] 1.
Proof: Let {v, [v.sub.1], [v.sub.2], ..., [v.sub.4n-1]} be the
vertices of [K.sub.1,4n-1] and {[e.sub.1], [e.sub.2], [e.sub.3], ...,
[e.sub.4n-1]} be the edges of denoted as in fig.14. We first label the
edges of [K.sub.1,4n-1] as follows f([e.sub.i]) = i for i = 1, 2, ...,
4n-1. Then the induced vertex labels are [f.sup.+]([v.sub.i]) = i for i
= 1, 2, ..., 4n-1; [f.sup.+](v) = 6n. Then the induced vertex labels are
all distinct. Hence, [K.sub.1,4n-1] is strong edge graceful for all n
[greater than or equal to] 1. The strong edge graceful labeling of
[K.sub.1,7] and [K.sub.1,11] are illustrated in fig.17 and fig 18
respectively.
[FIGURE 17 OMITTED]
[FIGURE 18 OMITTED]
Theorem 2.8: [K.sub.1,4n-1] is strong edge graceful for all n
[greater than or equal to] 1.
Proof: Let {v, [v.sub.1], [v.sub.2], ..., [v.sub.4n-1]} be the
vertices of [K.sub.1,4n-1] and {[e.sub.1], [e.sub.2], [e.sub.3], ...,
[e.sub.4n-1]} be the edges of [K.sub.1,4n-1] denoted as in fig.14. We
first label the edges of as follows f([e.sub.i]) = i for i = 1, 2, ...,
4n ; f([e.sub.4n+1]) = [3q/2] Then the induced vertex labels are
[f.sup.+]([v.sub.i]) = i for i = 1, 2, ..., 4n - 1 ;
[f.sup.+]([v.sub.4n+1]) = [3q/2] ; [f.sup.+](v) = 4n + 1 Then the
induced vertex labels are all distinct. Hence, [K.sub.1,4n-1] is strong
edge graceful for all n [greater than or equal to] 1. The strong edge
graceful labeling of [K.sub.1,5] and [K.sub.1,9] are illustrated in
fig.19 and fig. 20 respectively.
[FIGURE 19 OMITTED]
[FIGURE 20 OMITTED]
Definition 2.1: Let [P.sub.n] denote the path with n vertices. Then
the join of [K.sub.1] with [P.sub.n] is defined as fan and is denoted by
[F.sub.n] i.e. [F.sub.n] = [K.sub.1] + [P.sub.n].
Theorem 2.9: [F.sub.4n-2] is strong edge graceful graph for all n
[greater than or equal to] 1.
Proof : Let {v, [v.sub.1], [v.sub.2], [v.sub.3], ..., [v.sub.4n-2]}
be the vertices of [F.sub.4n-2] and the edges of [F.sub.4n-2] are
defined as follows as denoted in fig. 21.
[FIGURE 21 OMITTED]
We first label the edges of [F.sub.4n-2] as follows: f([e.sub.i]) =
i for i = 1 to 4n - 3, f(v[v.sub.i]) = 2n - i for i = 1 to 4n - 2 then
the induced vertex labels are [f.sup.+]([v.sub.1]) = 2(4n - 2);
[f.sup.+]([v.sub.i]) = f([v.sub.i-1]) + 1 for i = 2 to 4n - 1.;
[f.sup.+]([v.sub.n-2]) = 8n - 5; [f.sup.+](v) = 4n + 1
The labeling of [K.sub.6], [K.sub.10] are illustrated in fig.22 and
fig.23 respectively.
[FIGURE 22 OMITTED]
[FIGURE 23 OMITTED]
Theorem 2.10: [F.sub.4n+1] is strong edge graceful graph for all n
[greater than or equal to] 1.
Proof Let {v, [v.sub.1], [v.sub.2], [v.sub.3], ..., [v.sub.4n+1]}
be the vertices of [F.sub.4n+1] and the edges of [F.sub.4n+1] are
defined as follows {v[v.sub.i]: i = 1 to 4n + 1}[union] {[e.sub.i] =
([v.sub.i],[v.sub.i+1]): i = 1 to 4n} as denoted in fig21. We first
label the edges of [F.sub.4n+1] as follows: f([v.sub.i]) = i for i = 1
to 4n; f(v[v.sub.1]) = (4n + 1) f(v[v.sub.i]) = p (4n + 1) - i for i = 2
to (4n + 1) then the induced vertex labels are [f.sup.+]([v.sub.1]) = 4n
+ 2 ; [f.sup.+]([v.sub.i]) = n - 2 for i = 2 to 4n.
[f.sup.+]([v.sub.4n+1]) = 2n ; [f.sup.+](v) = 3n + 7/2
The labeling of [F.sub.5] are illustrated in fig.24..
[FIGURE 24 OMITTED]
REFERENCES
[1.] Harary, F. 1972. Graph Theory. Addison Wesley, Mass Reading
(1972).
[2.] Joseph, A. Gallian. 2007.A Dynamic Survey of Graph Labeling.
The Electronic journal of combinatorics 14(2007)#DS6.
[3.] Lo, S. 1985. On edge graceful labeling of graphs, Congressus
numerantium 50(1985).
[4.] Slamet, S. and Sugeng, K.A. 2008. Sharing scheme using magic
covering--Preprint. 2008.
B. Gayathri * and M. Subbiah@
PG and Research Dept. of Mathematics, Periyar E.V.R. College,
Trichy--23, India Email; maduraigayathri@gmail.com, mdthrcp@gmail.com