International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Impact Factor (2012): 3.358
Smooth Graceful Graphs And Its Application To
Construct Graceful Graphs
V J Kaneria1, M M Jariya2
1
Department of Mathematics, Saurashtra University, Rajkot, India
2
V. V. P. Engineering College, Rajkot, India
Abstract: In this paper we define smooth graceful labeling and we prove that cycle
Cn , complete bipartite graph K 2,n
are smooth graceful graphs. Using this smooth graceful labeling we prove that a graph obtained by joining
Cn ( n ≡ 0
4 )) and Wn
(mod
C
+
m
(m
≡2
and path
(mod
Pn
4 )) and
4 )) with a path of arbitrary length is graceful. We also prove a graph obtained by joining Cm ( m ≡ 0
(mod
with a path of arbitrary length is graceful.
Keywords: Graceful labeling, smooth graceful labeling, cycle, path, wheel and complete bipartite graph.
AMS subject classification number : 05C78.
1. Introduction
1979 , 219 − 229) proved that wheels
Wn = Cn + K1 are graceful. For detail survey of graph
labeling one can refer Gallian (Gallian, 2013 ).
(Frucht,
G = (V , E ) be a simple, undirected graph of size
( p, q ) i.e. | V |= p , | E |= q . For all standard
Let
terminology and notations we follow Harary (Harary
1972 ). We will give brief summery of definitions which
are used in this paper.
In this paper we introduce smooth graceful labeling and
using this we prove that a graph obtained by joining
( m ≡ 2 (mod 4 )) and
Cm+
Cn ( n ≡ 0 (mod 4 )) with a
Definition − 1.1 : If the vertices of the graph are assigned
values subject to certain conditions then it is known as
graph labeling.
path is graceful. We also prove a graph obtained by
Most interesting graph labeling problems have three
important ingrediants:
Definition − 1.3 : A bipartite graceful graph G with
graceful labeling f is said to be smooth graceful graph
if
it
admits
an
injective
map
• A set of numbers from which vertex labels are chosen.
• A rule that assigns a value to each edge.
• A condition that these values must satisfies.
Definition − 1.2 :
labeling
of
A function
a
f : V → {0,1, , q}
graph
f is called graceful
G = (V , E )
if
is injective and the induce
f å : E → {1,2,, q} defined as f å (e)
= | f (u ) − f (v) | is bijective for every edge
e = (u , v) ∈ E . A graph G is called graceful graph if it
function
admits a graceful labeling.
The graceful labeling was introduced by A. Rosa (Rosa,
1967 , p. 349 − 355). He proved that cycle Cn is
n ≡ 0,3 (mod 4 ). Bloom and Golumb
(Bloom and Golomb, 1978 , p. 53 − 65) proved that the
complete graph K n is graceful, when n ≤ 4 . Hoede
and Kuiper (Hoede and Kuiper, 1987 , 311), Frucht
graceful, when
joining
Cm ( m ≡ 0 (mod 4 )) and Wn with a path is
graceful.
g : V → {0,1, , 
q −1 q + 1
q+3
, 
 + l, 
+l
2
2
2
, q + l} such that its induce edge labeling map
g å : E → {1 + l ,2 + l ,, q + l}
defined
as
å
g (e) =| g (u ) − g (v) | , for every edge e = (u , v) ∈ E
, for any l ∈ N is a bijection.
Smooth graceful graph will help to produce new
disconnected as well as connected graceful graphs.
2. Main Results
Theorem − 2.1 : A cycle
Cn , ( n ≡ 0 (mod 4 )) is a
smooth graceful graph.
Proof : Let v1 , v2 , , vn be vertices of
Cn . We know
Cn ( n ≡ 0 (mod 4 )) is a bipartite graph and
f : V (Cn ) → {0,1,, n} defined by
that
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Paper ID: 02015488
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909
International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Impact Factor (2012): 3.358
i −1
, ∀ i = 1,3,  , n − 1 .
2
i−2
n
= q−(
) , ∀ i = 2,4,,
2
2
i
n
n
= q − ( ) , ∀ i = + 2, + 4,, n is graceful
2
2
2
labeling for Cn , when n ≡ 0 (mod 4 ).
f (vi ) =
Now
we
define
n −1 n +1
n+3
, 
 + l, 
 + l ,  , n + l}
g : V (Cn ) → {0,1,, 
2
2
2
such
that
its
induce
edge
labeling
map
g å : E (Cn ) → {1 + l ,2 + l , , q + l} defined by
g (u ) = f (u ) , ∀ u ∈{v1 , v3 , , vn −1}
= f (u ) + l ,
.
by taking
k ∈ {1,2,  , n} , ek = (vk , vk +1 ) ∈ E ,
vn+1 = v1 ,
å
g (ek ) =| g (vk − g (vk +1 ) |
= f (ui ) − f (v j ) | +l
= f å ( e) + l .
å
So g ( E ) = {1 + l ,2 + l , ,2n + l} . Therefore
a bijection.
Hence
g å is
K 2,n is a smooth graceful graph.
Pn of length n − 1 is a
smooth graceful graph, ∀ n ∈ N .
Theorem − 2.3 : A path
v1 , v2 ,, vn be vertices of Pn . We know
Pn is a bipartite and graceful graph with graceful
labeling f : V ( Pn ) → {0,1, , n − 1} defined by
i−2
f (vi ) =
, if i ≡ 0 (mod 2 )
2
i +1
= n−(
) , if i ≡ 1 (mod 2 ), ∀ i = 1,2,  , n .
2
that
Now
we
define
n−2 n
n+2
,   + l , 
 + l ,  , n − 1 + l}
g : V ( Pn ) → {0,1,, 
2
2
2
and
its
induce
edge
labeling
map
=| f (vk − f (vk +1 ) | +l
= f å (ek ) + l , so g å ( E ) = {1 + l ,2 + l ,, n + l} .
Therefore
e = (ui , v j ) ∈ E ( i = 1,2,1 ≤ j ≤ n ),
g å (e) =| g (ui ) − g (v j ) |
Proof : Let
∀ u ∈{v2 , v4 ,, vn } and
g å (e) =| g (u ) − g (v) | , for every edge e = (u , v) ∈ E
Since for any
Since for any
g å is a bijective map. Hence Cn , ( n ≡ 0
(mod 4 )) is a smooth graceful graph.
g å : E ( Pn ) → {1 + l ,2 + l , , n − 1 + l} defined by
g (v) = f (v ) + l , if i ≡ 1 (mod 2 )
= f (v) , if i ≡ 0 (mod 2 ), ∀ i = 1,2,  , n
Theorem − 2.2 : A complete bipartite graph
K 2,n is a
and
smooth graceful graph.
Proof : Let u1 , u2 , v1 , v2 , , vn be vertices of
K 2,n .
Since for any
g å (e) =| g (u ) − g (v) | , for every e = (u , v) ∈ E .
e = (vi , vi +1 ) ∈ E ( 1 ≤ i ≤ n − 1 ),
å
We know that
f : V ( K 2, n ) → {0,1,  ,2n} defined
by
f (vi ) = i − 1 , ∀ i = 1,2,  , n .
f (ui ) = in , ∀ i = 1,2
labeling K 2,n .
Now
is a graceful
we
its
induce
define
edge
= | f (vi ) − f (vi +1 ) | +l
=
f å ( e) + l
,
we
å
g ( E ) = {1 + l ,2 + l ,, n − 1 + l} .
shall
labeling
map
g å : E ( K 2,n ) → {1 + l ,2 + l ,  , n + l} defined by
g (u ) = f (u ) + l , ∀ u ∈ {u1 , u 2 }
= f (v) , ∀ v ∈{v1 , v2 ,, vn } and g å (e)
| g (u ) − g (v) | , ∀ e = (u , v) ∈ E .
g å is a bijection and so Pn of length n − 1
is a smooth graceful, ∀ n ∈ N .
Theorem − 2.4 : Let
Cm+ ( m ≡ 2 (mod 4 )) be a cycle
with twin chords, they form a triangle with an edge of the
Cm . Then the graph obtained by joining Cm+ and
a cycle Cn ( n ≡ 0 (mod 4 )) with a path Pt +1 of
arbitrary length t is graceful.
cycle
=
Volume 3 Issue 8, August 2014
Paper ID: 02015488
have
Therefore
g : V ( K 2,n ) → {0,1,  , n − 1, n + l , n + 1 + l ,  , n + l}
and
g (e) =| g (vi ) − g (vi +1 ) |
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Proof : Let G be a graph obtained by joining
Cm+ and
Cn with Pt +1 , where Cm+ is a cycle with twin chords
and they form a triangle with an edge of the cycle Cm ,
Cn is a cycle on n vertices and Pt +1 is a path on
t + 1 vertices. Let u1 , u2 ,, um be vertices of Cm+
(u2 , um ), (u2 , um−1 )
v1 = um− 2 , v2 ,, vt +1 be vertices of the path Pt +1 of
t length and w1 = vt +1 , w2 ,, wn be vertices of the
cycle Cn .
with
twin
chords
We define labeling function
where
f : V (G ) → {0,1,  , q} ,
q = m + t + n + 2 as follows:
f (u1 ) = q − 1 , f (um ) = q , f (um −1 ) = q − 2 ,
j−2
m
f (u j ) =
, ∀ j = 2,4,  , − 1
2
2
j
m
m
=
, ∀ j=
+ 1, + 3,, m − 2
2
2
2
j +3
= q−(
) , ∀ j = 3,5 , m − 3 ;
2
m+i
f (vi ) = q − (
) , when i ≡ 0 (mod 2 )
2
m+i−3
=
, when i ≡ 1 (mod 2 ), ∀
2
i = 2,3,  , t + 1 ;
m+t + k −3
f ( wk ) =
, ∀ k = 3,5,  , n − 1
2
and t is even or
n
∀ k = 2,4,, and t is odd
2
m+t +k
= q−(
) , ∀ k = 3,5,  , n − 1 and t is
2
odd or
n
and t is even
2
m+t +k +2
n
n
= q−(
) , ∀ k = + 2, + 4,, n
2
2
2
and t is even
m + t + k −1
n
n
=
, ∀ k = + 2, + 4,  , n and t
2
2
2
∀ k = 2,4,,
is odd.
Above labeling pattern give rise graceful labeling to the
given graph G .
Illustration − 2.5 : A cycle
P9 of 8 length with its graceful
labeling shown in figure − 1 .
joining by a path
Theorem − 2.6 : A graph obtained by joining
m ≡ 0 (mod 4 )) and a wheel Wn = Cn + K1 with
a path of arbitrary length t is graceful.
Proof : Let G be a graph obtained by joining
Cm and
Wn with Pt +1 a path of length t on t + 1 vertices.
u1 , u2 ,, um
Cm ,
Let
be
vertices
of
v1 = um , v2 ,, vt +1 be vertices of path Pt +1 of t
length and w0 , w1 , , wn = vt +1 be vertices of the
wheel Wn in which w0 is adjacent with each wi (
1 ≤ i ≤ n ).
We know that wheel
Wn is a graceful graph with
labeling function f1 defined as follows:
f1 ( w0 ) = 0 , f1 ( wn ) = 2n
f1 ( wi ) = 2 n − i − 1 , if i is even ( 2 ≤ i ≤ n − 2 )
and n is even
= 2 , if i = n − 1 and n is even
= i , if i is odd ( 1 ≤ i ≤ n − 3 ) and n is even
= 2i , if i = 1 and n is odd
= n + i , if i is even ( 2 ≤ i ≤ n − 1 ) and n is odd
= n +1 − i , if i is odd ( 3 ≤ i ≤ n − 2 ) and n is
odd.
We define function
f : V (G ) → {0,1,  , q} , where
q = m + t + 2n as follows:
j −1
f (u j ) = q − (
) , ∀ j = 1,3, , m − 1 .
2
j−2
m
=
, ∀ j = 2,4,  ,
2
2
j
m
m
=
, ∀ j=
+ 2, + 4,, m ;
2
2
2
m+i
f (vi ) = q + 1 − (
) , when i ≡ 0 (mod 2 )
2
m + i −1
=
, when i ≡ 1 (mod 2 ), ∀
2
i = 2,3,  , t ;
m+t
f ( wk ) = 2n + (
) − f1 ( wk ) , when t is even
2
m + t +1
f1 ( wk ) +
=
, when t is odd, ∀
2
k = 0,1, , n .
C10+ with twin chords, C12
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Above labeling pattern give rise graceful labeling to the
given graph G .
Illustration − 2.7 : A cycle C12 , a wheel
W7 joining
P6 of 5 length with its graceful labeling
shown in figure − 2 .
by a path
3. Concluding Remarks
We have introduced a new graph labeling is called smooth
graceful labeling. We have proved that Cn n ≡ 0
(mod 4 ),
K 2,n , Pn are smooth graceful graphs. Using
Figure 2: A cycle C12 , a wheel
W7 joining by a path
P6 of 5 length with its graceful labeling
these we have got graceful labeling for two new families
of graphs. Present work contribute five new results. The
labeling pattern is demonstrated by illustrations.
References
[1] G. S. Bloom and S. W. Golomb, Numbered complete
graphs, usual rules and assorted applications, in Theory
and applications of Graphs, Lecture Notes in Math.,
642 , Springer − Verlag, New York (1978)
53 − 65 .
[2] J. A. Gallian, The Electronics Journal of
Combinatorics, 16, # DS6(2013).
[3] R. Frucht, Graceful numbering of wheels and related
graphs, Ann. N. Y. Acad. Sci., 319 (1979) 219 − 229.
[4] F. Harary, Graph theory Addition Wesley,
Massachusetts, 1972 .
[5] C. Hoede and H. Kuiper, All wheels are graceful, Util.
Math., 14 (1987) 311 .
[6] A. Rosa, On certain valuation of graph, Theory of
Graphs (Rome, July 1966), Goden and Breach, N. Y.
and Paris, 1967, 349 − 355.
Figure 1: A cycle
a path
C10+ with twin chords, C12 joining by
P9 of 8 length and its graceful labeling
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