ISSN 2319-8133 (Online) (An International Research Journal), www.compmath-journal.org
Fuzzy 10
k- Based Graceful Labeling for Graphs Obtained from Circuits Merged with Paths
A. Solairaju
1and T. Narppasalai Arasu*
21
Associate Professor of Mathematics,
Jamal Mohamed College (Autonomous), Trichy, INDIA.
2
Assistant Professor of Mathematics,
Chikkanna Government Arts College Tirupur, INDIA.
email :
1[email protected],
2[email protected]
(Received on: September 6, 2018)
ABSTRACT
A labeling graph G which can be gracefully numbered is called graceful. A graph which admits a fuzzy 10k-based graceful labeling is called a fuzzy 10k-based graceful graph. In this paper the existence of fuzzy 10k- based gracefulness to graphs obtained from circuits merged with paths were discussed.
Mathematical Subject Classification 2010: 05C78, 05C72.
Keywords: Graceful labeling, Fuzzy graceful labeling, Fuzzy 10k-based graceful labeling, Mirror image of a graph.
1. INTRODUCTION
Graph theory is an important tool to represent real world problems. Rosenfeld
introduced the concept of fuzzy graphs since it is not always possible to represent all systems
in conventional graph theory due to the uncertainty of the parameters in the system. Fuzzy
relation on a set was defined by Zadeh
8in 1965 and definition of a fuzzy graph was given by
Rosenfeld and kauffmann in 1973. The concept of graceful labeling has been introduced by
Rosa
6in 1976. A. Nagoorgani, D. Rajalakshmi(a)Subhasini
4,5and Bhattacharya P
1discussed
some properties of graph labeling using fuzzy graph theory. R. Jebesty Shajila and S.Vimala
3performed fuzzy vertex labeling on wheel and fan graphs . The graphs which are used in this
paper are discussed in the dynamic survey of graph labeling by J.A. Gallian
2and the graphs
introduced by Vaidya S.K and Lekha Bijukumar
7. In this paper we discuss the fuzzy 10
k-based
graceful labeling for nC *P , nP *P ,mirror image of nP *P and nC *2P *(n-1)P graphs.
Definition 2.1: A path P
nwith n vertices is a walk in which no vertex appears more than once.
Definition 2.2 : A closed path P
nis called a cycle C
n.
Definition 2.3 : A fuzzy graph G = 〈 , 〉 is a pair of functions : → [0,1] and : × → [0,1] such that for all , ∈ , ℎ ( , ) ≤ ( )⋀ ( )
Definition 2.4 : A labeling of a graph is an assignment of values to the vertices and edges of a graph.
Definition2. 5 : A graceful labeling of a graph G with q edges is an injection : ( ) → {0,1,2, … } such that when each edge is assigned the distinct label | ( ) − ( )|
Definition2. 6 : A graph G = 〈 , 〉 is said to be a fuzzy labeling if : → [0,1] and
: × → [0,1] is bijective such that the membership value of edges and vertices are distinct and ( , ) < ( )⋀ ( ) for all , ∈
3. MAIN RESULTS
Definition 3.1 : A fuzzy graph G = 〈 , 〉 with p vertices and q edges is said to be a fuzzy 10
k-based graceful labeling if k is a least positive integer such that 0 < + < 10 and : → [0, ] and : × → [0,1]are injective such that the membership value of edges and vertices are distinct and satisfying the following four conditions
(i) ( , ) < ( )⋀ ( ) for all , ∈ (ii) ( , ) = | ( ) − ( )| for all , ∈
(iii) { ( ) × 10 ∈ } ⊂ { + 1, + 2, … 2 + 1}
(iv) { ( ) × 10 ∈ } = {1,2, … }
Definition 3.2 : The graph nC
4*2P
nis a connected graph whose vertex set is { , , … } and edge set is
{ , : = 1 − 1} ∪ { , , , : = 1 }
Here p = 4n and q= 6n-2.
Theorem 3.3: The graph = nC ∗ 2P is a Fuzzy 10
k-based graceful graph.
Proof : Consider a graph G = 〈 , 〉 with p =4n vertices { , , … } and q = 6n-2 edges
{ , : = 1 − 1} ∪ { , , , : = 1 }
such that
: → [0,1] and : × → [0,1] defined by ( ) = 2 + 1
10
(
)( ) − − 5
10
( ) = ( ) − 6
10 = 3,5, … − 1
( ) = ( ) + 6
10 = 4,6, … − 1
( ) = ( ) − 1
10
( ) = ( ) − 2
10
( ) =
( )− 6
10 = 3,5, … − 1
( ) =
( )+ 6
10 = 4,6, … − 1 ( ) = ( ) − 10
( ) = ( ) − − 2
10
( ) =
( )+ 6
10 = 3,5, … − 1
( ) =
( )+ 6
10 = 3,5, … − 1
( ) = ( ) − 3
10
( ) =
( )− 6
10 = 4,6, … − 1
( ) = ( ) − 4
10
( ) =
( )− 6
10 = 4,6, … − 1
For example the fuzzy 10
k-based graceful labelling of, 5C ∗ 2P and 6C ∗ 2P are shown in Fig.3. 1 and Fig.3.2
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Fig.3.2
Fig.3.3
Definition 3.4 : The graph nP ∗ P is a connected graph containing n copies of P
3and a copy of P
2nwhose vertex set is { , , … } and edge set is
{ : = 1 2 − 1} ∪ { , : = 1 } Here p = 3n and q= 4n-1.
Theorem 3.5 : The graph = nP ∗ P is a Fuzzy 10
k-based graceful graph.
Proof : Consider a graph G = 〈 , 〉 with p =3n vertices { , , … } and q = 4n-1 edges
{ : = 1 2 − 1} ∪ { , : = 1 } such that
: → [0,1] and : × → [0,1] defined by
( ) = 2 + 1
10
( ) = ( ) − 10
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( ) = ( ) − 1 10 ( ) = ( ) + 2
10 = 3,5, … 2 − 1 ( ) = ( ) − 2
10 = 4,6, … 2
( ) = ( ) + 1
10
( ) =
( )− 2
10 = 3,5, . . − 1
( ) =
( )+ 2
10 = 4,6, . . − 1
For example the fuzzy 10
k-based graceful labelling of , 6P ∗ P is shown in Fig.3.3.
Definition 3.6 : The mirror image of the graph nP ∗ P is a connected graph containing two times of n copies of P
3and a copy of P
2nwhose vertex set is {v , v , … v } and edge set is {v v : i = 1 to 2n − 1} ∪ {v v , v v : i = 1 to n} ∪ {v v , v v : i = 1 to n} .Here p = 4n and q= 6n-1.
Theorem 3.7: The mirror image of the graph G = nP ∗ P is a Fuzzy 10
k-based graceful graph.
Proof : Consider a graph G = 〈σ, μ〉 with p =4n vertices {v , v , … v } and q = 6n-1 edges {v v : i = 1 to 2n − 1} ∪ {v v , v v : i = 1 to n} ∪ {v v , v v : i = 1 to n}
such that
σ: V → [0,1] and μ: V × V → [0,1] defined by σ(v ) = 2q + 1
10 σ(v ) = σ(v ) − q
10 σ(v ) = σ(v ) − 2
10 σ v
( )= σ(v ) + 6
10 for i = n, n − 2, … 1 or 2 if n is odd or even σ(v ) = σ(v ) − 3
10 σ v
( )= σ(v ) − 6
10 for i = n − 1, n − 3, … 2 or 1 if n is odd or even σ(v ) = σ(v ) − q − 1
10 σ v
( )= σ(v ) + 6
for i = n + 1, n + 3, … 2n or 2n − 1 if n is odd or even
σ(v ) = σ(v ) − 10 σ v
( )= σ(v ) − 6
10 for i = n + 2, n + 4, … 2n − 1 or 2n if n is odd or even σ(v ) = σ(v ) − 6
10 for i = 3n, 3n − 2, . .2n + 1 or 2n + 2 if n is odd or even σ(v ) = σ(v ) + 3
10 σ(v ) = σ(v ) + 6
10 for i = 3n − 1,3n − 3, . .2n + 2 or 2n + 1 if n is odd or even σ(v ) = σ(v ) − 6
10 for i = 4n, 4n − 2, . .3n + 1 or 3n + 2 if n is odd or even σ(v ) = σ(v ) + 4
10 σ(v ) = σ(v ) + 6
10 for i = 4n − 1,4n − 3, . .3n + 2 or 3n + 1 if n is odd or even For example the mirror image of the fuzzy 10
k-based graceful labeling of 6P ∗ P is shown in Fig.3.4.
Fig.3.4
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Definition 3.8 : The graph nC ∗ 2P ∗ (n − 1)P is defined as a connected graph whose vertex set is {v , v , … v } and edge set is
{v v , v v : i = 1 to n − 1} ∪ {v v , v v , v v , v v : i = 1 to n} ∪ {v v : i = 1 to n − 1} Here p = 4n and q = 7n-3.
Theorem 3.9 : The graph G = nC ∗ 2P ∗ (n − 1)P is a Fuzzy 10
k-based graceful graph.
Proof : Consider a graph G = 〈σ, μ〉 with p =4n vertices {v , v , … v } and q = 7n-3 edges {v v , v v : i = 1 to n − 1} ∪ {v v , v v , v v , v v : i = 1 to n} ∪ {v v : i = 1 to n − 1} such that
σ: V → [0,1] and μ: V × V → [0,1] defined by σ(v ) = 2q + 1
10
σ(v
)σ(v ) − q − 4 10 σ(v ) = σ(v ) − 7
10 for i = 3,5, … n or n − 1 if n is odd or even σ(v ) = σ(v ) + 7
10 for i = 4,6, … n − 1 or n if n is odd or even σ(v ) = σ(v ) − 1
10 σ(v ) = σ(v ) + 1 10 σ(v ) = σ v
( )− 7
10 for i = 3,5, … n or n − 1 if n is odd or even σ(v ) = σ v
( )+ 7
10 for i = 4,6, … n − 1 or n if n is odd or even σ(v ) = σ(v ) − q
10 σ(v ) = σ(v ) − q − 2
10 σ(v ) = σ v
( )+ 7
10 for i = 3,5, … n or n − 1 if n is odd or even σ(v ) = σ v
( )+ 7
10 for i = 3,5, … n or n − 1 if n is odd or even σ(v ) = σ(v ) − 3
10 σ(v ) = σ v
( )− 7
10 for i = 4,6, … n − 1 or n if n is odd or even σ(v ) = σ(v ) − 5
10 σ(v ) = σ v
( )− 7
10 for i = 4,6, … n − 1 or n if n is odd or even
of 4C ∗ 2P ∗ 3P and of 5C ∗ 2P ∗ 4P are shown in Fig.3.5 and Fig.3.6
Fig.3.5
Fig.3.6
4. CONCLUSION
In this paper the concept of fuzzy 10
k-based gracefulness for graphs obtained from circuits merged with paths has been introduced . The extension of this study on some more special graphs is under process.
REFERENCES
1. Bhattacharya P. Some remarks on fuzzy graphs, Pattern Recognition Letter, Vol 6, No.
5,297-302 (1987).
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