Mathematical Combinatorics. An International Book Series, vol. 1, 2011

2011·1.2 MB·English

by  Mao, Linfan.

ISBN 978-1-59973-146-9
VOLUME 1, 2011
MATHEMATICAL COMBINATORICS
(INTERNATIONAL BOOK SERIES)
Edited By Linfan MAO
THE MADIS OF CHINESE ACADEMY OF SCIENCES
March, 2011
Vol.1, 2011 ISBN 978-1-59973-146-9
Mathematical Combinatorics
(International Book Series)
Edited By Linfan MAO
The Madis of Chinese Academy of Sciences
March, 2011
Aims and Scope: The Mathematical Combinatorics (International Book Series)
(ISBN978-1-59973-146-9)isafullyrefereedinternationalbookseries,sponsoredbytheMADIS
of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 pages
approx. per volume, which publishes original research papers and survey articles in all as-
pects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics,
non-euclidean geometry and topology and their applications to other sciences. Topics in detail
to be covered are:
Smarandache multi-spaces with applications to other sciences, such as those of algebraic
multi-systems, multi-metric spaces, , etc.. Smarandache geometries;
···
Differential Geometry; Geometry on manifolds;
Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and
map enumeration; Combinatorial designs; Combinatorial enumeration;
Low Dimensional Topology; Differential Topology; Topology of Manifolds;
Geometrical aspects of Mathematical Physics and Relations with Manifold Topology;
Applications of Smarandache multi-spaces to theoretical physics; Applications of Combi-
natorics to mathematics and theoretical physics;
Mathematical theory on gravitationalfields; Mathematical theory on parallel universes;
Other applications of Smarandache multi-space and combinatorics.
Generally, papers on mathematics with its applications not including in above topics are
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Mansfield University, USA
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M.Khoshnevisan
Linfan MAO
School of Accounting and Finance,
Chinese AcademyofMathematicsandSystem
Griffith University, Australia
Science, P.R.China
Email: [email protected] Xueliang Li
Nankai University, P.R.China
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Editors
Han Ren
East China Normal University, P.R.China
S.Bhattacharya
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Deakin University
Geelong Campus at Waurn Ponds W.B.Vasantha Kandasamy
Indian Institute of Technology, India
Australia
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Email: [email protected]
Mingyao Xu
An Chang
Peking University, P.R.China
Fuzhou University, P.R.China
Email: [email protected]
Email: [email protected]
Guiying Yan
Junliang Cai
Chinese AcademyofMathematicsandSystem
Beijing Normal University, P.R.China
Science, P.R.China
Email: [email protected]
Email: [email protected]
Yanxun Chang
Y. Zhang
Beijing Jiaotong University, P.R.China
Department of Computer Science
Email: [email protected] Georgia State University, Atlanta, USA
Shaofei Du
Capital Normal University, P.R.China
Email: [email protected]
Florentin Popescu and Marian Popescu
University of Craiova
Craiova, Romania
Xiaodong Hu
Chinese AcademyofMathematicsandSystem
Science, P.R.China
Email: [email protected]
Yuanqiu Huang
Hunan Normal University, P.R.China
Email: [email protected]
Achievement provides the only real pleasure in life.
By Thomas Edison, an American inventor.
Math.Combin.Book Ser. Vol.1 (2011), 01-19
Lucas Graceful Labeling for Some Graphs
M.A.Perumal1, S.Navaneethakrishnan2 and A.Nagarajan2
1. DepartmentofMathematics,NationalEngineeringCollege,
K.R.Nagar,Kovilpatti,TamilNadu,India
2. DepartmentofMathematics,V.O.CCollege,Thoothukudi,TamilNadu,India.
Email: [email protected],[email protected],[email protected]
Abstract: A Smarandache-Fibonacci triple is a sequence S(n), n 0 such that
≥
S(n) = S(n 1) + S(n 2), where S(n) is the Smarandache function for integers
− −
n 0. Clearly, it is a generalization of Fibonacci sequence and Lucas sequence. Let
≥
G be a (p,q)-graph and S(n)n 0 a Smarandache-Fibonacci triple. An bijection
{ | ≥ }
f: V(G) S(0),S(1),S(2),...,S(q) is said to be a super Smarandache-Fibonacci grace-
→{ }
ful graph if the induced edge labeling f∗(uv) = f(u) f(v) is a bijection onto the set
| − |
S(1),S(2),...,S(q) . Particularly, if S(n),n 0 is just the Lucas sequence, such a label-
{ } ≥
ing f :V(G) l0,l1,l2, ,la (a ǫN) is said to beLucas graceful labelingif theinduced
→{ ··· }
edge labeling f1(uv) = f(u) f(v) is a bijection on to the set l1,l2, ,lq . Then G is
| − | { ··· }
called Lucas graceful graph if it admits Lucas graceful labeling. Also an injective function
f :V(G) l0,l1,l2, ,lq issaid tobestrongLucasgracefullabelingiftheinducededge
→{ ··· }
labeling f1(uv) = f(u) f(v) is a bijection onto the set l1,l2,...,lq . G is called strong
| − | { }
Lucas graceful graph if it admits strong Lucas graceful labeling. In this paper, we show
that some graphs namely Pn, Pn+−e, Sm,n, Fm@Pn, Cm@Pn, K1,n⊙2Pm, C3@2Pn and
Cn@K1,2 admitLucasgraceful labeling andsome graphsnamely K1,n and Fn admitstrong
Lucas graceful labeling.
KeyWords: Smarandache-Fibonaccitriple,superSmarandache-Fibonaccigracefulgraph,
Lucas graceful labeling, strong Lucas graceful labeling.
AMS(2010): 05C78
§1. Introduction
By a graph, we mean a finite undirected graph without loops or multiple edges. A path of
length n is denoted by P . A cycle of length n is denoted by C .G+ is a graph obtained from
n n
thegraphGbyattachingapendantvertextoeachvertexofG. Theconceptofgracefullabeling
was introduced by Rosa [3] in 1967.
A function f is a graceful labeling of a graph G with q edges if f is an injection from
1ReceivedNovember11,2010. Accepted February10,2011.
2 M.A.Perumal,S.NavaneethakrishnanandA.Nagarajan
the vertices of G to the set 1,2,3, ,q such that when each edge uv is assigned the la-
{ ··· }
bel f(u) f(v), the resulting edge labels are distinct. The notion of Fibonacci graceful
| − |
labeling was introduced by K.M.Kathiresan and S.Amutha [4]. We call a function, a Fi-
bonacci graceful labeling of a graph G with q edges if f is an injection from the vertices of
G to the set 0,1,2,...,F , where F is the qth Fibonacci number of the Fibonacci series
q q
{ }
F = 1,F = 2,F = 3,F = 5,..., and each edge uv is assigned the label f(u) f(v). Based
1 2 3 4
| − |
on the above concepts we define the following.
Let G be a (p,q) -graph. An injective function f : V(G) l ,l ,l , ,l , (a ǫ N),
0 1 2 a
→ { ··· }
is said to be Lucas graceful labeling if an induced edge labeling f (uv) = f(u) f(v) is a
1
| − |
bijection onto the set l ,l , ,l with the assumption of l = 0,l = 1,l = 3,l = 4,l =
1 2 q 0 1 2 3 4
{ ··· }
7,l =11, ,. ThenGiscalledLucasgracefulgraphifitadmitsLucasgracefullabeling. Also
5
···
aninjectivefunctionf :V(G) l ,l ,l , ,l issaidtobestrongLucasgracefullabelingif
0 1 2 q
→{ ··· }
theinducededgelabelingf (uv)= f(u) f(v) isabijectionontotheset l ,l , ,l . Then
1 1 2 q
| − | { ··· }
GiscalledstrongLucasgracefulgraphifitadmitsstrongLucasgracefullabeling. Inthispaper,
weshowthatsomegraphsnamelyP , P+ e, S , F @P , C @P , K 2P , C @2P
n n − m,n m n m n 1,n⊙ m 3 n
andC @K admitLucasgracefullabelingandsomegraphsnamelyK andF admitstrong
n 1,2 1,n n
Lucas graceful labeling. Generally, let S(n), n 0 with S(n) = S(n 1)+S(n 2) be a
≥ − −
Smarandache-Fibonacci triple, where S(n) is the Smarandache function for integers n 0. An
≥
bijection f: V(G) S(0),S(1),S(2),...,S(q) is said to be a super Smarandache-Fibonacci
→ { }
graceful graph if the induced edge labeling f (uv) = f(u) f(v) is a bijection onto the set
∗
| − |
S(1),S(2), ,S(q) .
{ ··· }
§2. Lucas graceful graphs
In this section, we show that some well known graphs are Lucas graceful graphs.
Definition 2.1 Let G be a (p,q) -graph. An injective function f :V(G) l ,l ,l , ,l , ,
0 1 2 a
→{ ··· }
(a ǫ N) is said tobe Lucas graceful labeling if an induced edge labeling f (uv)= f(u) f(v) is
1
| − |
a bijection onto the set l ,l , ,l with the assumption of l = 0,l =1,l =3,l = 4,l =
1 2 q 0 1 2 3 4
{ ··· }
7,l =11, ,. Then G is called Lucas graceful graph if it admits Lucas graceful labeling.
5
···
Theorem 2.2 The path P is a Lucas graceful graph.
n
Proof Let P be apath oflengthn having(n+1)verticesnamely v ,v ,v , ,v ,v .
n 1 2 3 n n+1
···
Now, V(P ) = n+1 and E(P ) = n. Define f : V(P ) l ,l ,l , ,l , ,a ǫ N by
n n n 0 1 2 a
| | | | → { ··· }
f(u )=l ,1 i n. Next, we claim that the edge labels are distinct. Let
i i+1
≤ ≤
E = f (v v ):1 i n = f(v ) f(v ) :1 i n
1 i i+1 i i+1
{ ≤ ≤ } {| − | ≤ ≤ }
= f(v ) f(v ) , f(v ) f(v ) , , f(v ) f(v ) ,
1 2 2 3 n n+1
{| − | | − | ··· | − | }
= l l , l l , , l l = l ,l , ,l .
2 3 3 4 n+1 n+2 1 2 n
{| − | | − | ··· | − |} { ··· }
So, the edges of P receive the distinct labels. Therefore, f is a Lucas graceful labeling.
n
Hence, the path P is a Lucas graceful graph. (cid:3)
n
LucasGracefulLabelingforSomeGraphs 3
Example 2.3 The graph P admits Lucas graceful Labeling, such as those shown in Fig.1
6
following.
l l l l l l l
2 3 4 5 6 7 8
l l l l l l
1 2 3 4 5 6
Fig.1
Theorem 2.4 P+ e,(n 3) is a Lucas graceful graph.
n − ≥
Proof Let G=P+ e with V(G)= u ,u , ,u v ,v , ,v be the vertex
n − { 1 2 ··· n+1} { 2 3 ··· n+1}
set of G. So, V(G) = 2n+1 and E(G) = 2n. Define f :V(G) l ,l ,l , ,l , ,a ǫ N,
| | | | S → { 0 1 2 ··· a }
by
f(u )=l ,1 i n+1 and f(v )=l ,2 j n+1.
i 2i 1 j 2(j 1)
− ≤ ≤ − ≤ ≤
We claim that the edge labels are distinct. Let
E = f (u u ):1 i n = f(u ) f(u ) :1 i n
1 1 i i+1 i i+1
{ ≤ ≤ } {| − | ≤ ≤ }
= f(u ) f(u ), f(u ) f(u ), , f(u ) f(u )
1 2 2 3 n n+1
{| − | | − | ··· | − |}
= l l , l l , , l l = l ,l , ,l ,
1 3 3 5 2n 1 2n+1 2 4 2n
{| − | | − | ··· | − − |} { ··· }
E = f (u v ):2 i, j n
2 1 i j
{ ≤ ≤ }
= f(u ) f(v ), f(u ) f(v ), , f(u ) f(v )
2 2 3 3 n+1 n+1
{| − | | − | ··· | − |}
= l l , l l , , l l = l ,l , ,l .
3 2 5 4 2n+1 2n 1 3 2n 1
{| − | | − | ··· | − |} { ··· − }
Now, E =E E = l ,l , ,l ,l . So, the edges of G receive the distinct labels.
1 2 1 3 2n 1 2n
∪ { ··· − }
Therefore, f is a Lucas graceful labeling. Hence, P+ e,(n 3) is a Lucas graceful graph. (cid:3)
n − ≥
Example 2.5 The graphP+ e admits Lucas gracefullabeling, suchasthsoe shownin Fig.2.
8 −
l l l l l l l l l
1 3 5 7 9 11 13 15 17
l l l l l l l l
2 4 6 8 10 12 14 16
l l l l l l l l
1 3 5 7 9 11 13 15
l l l l l l l l
2 4 6 8 10 12 14 16
Fig.2
Definition 2.6([2]) Denote by S such a star with n spokes in which each spoke is a path of
m,n
length m.
Theorem 2.7 The graph S is a Lucas graceful graph when m is odd and n 1,2(mod 3).
m,n
≡
4 M.A.Perumal,S.NavaneethakrishnanandA.Nagarajan
Proof Let G=S and let V(G)= ui :1 i m and 1 j n be the vertex set of
m,n j ≤ ≤ ≤ ≤
S . Then V(G) = mn+1 and E(G) = mn. Define f : V(G) l ,l ,l , ,l , ,a ǫ N
m,n | | | |(cid:8) → { 0(cid:9)1 2 ··· a }
by
f(u )=l for i=1,2, ,m 2 and i 1(mod 2);
0 0
··· − ≡
f ui =l ,1 j n for i=1,2, ,m 1 and i 0(mod 2);
j n(i−1)+2j−1 ≤ ≤ ··· n− ≡
f(cid:0)uij(cid:1)=lni+2−2j,1≤j ≤n and for s=1,2,··· , 3,
f(cid:0)um(cid:1) =l ,3s 2 j 3s.
j n(m−1)+2(j+1)−3s − ≤ ≤
We clai(cid:0)m th(cid:1)at the edge labels are distinct. Let
m m
E = f u ui = f(u ) f ui
1 1 0 1 0 − 1
i 1([im=1od 2)(cid:8) (cid:0) (cid:1)(cid:9) i 1(i[m=1od 2)(cid:8)(cid:12) (cid:0) (cid:1)(cid:12)(cid:9)
≡ ≡ (cid:12) (cid:12)
m m
= l l = l
0 n(i 1)+1 n(i 1)+1
− − −
i 1([im=1od 2)(cid:8)(cid:12) (cid:12)(cid:9) i 1([im=1od 2)(cid:8) (cid:9)
≡ (cid:12) (cid:12) ≡
= l ,l ,l , ,l ,
1 2n+1 4n+1 n(m 1)+1
··· −
(cid:8) (cid:9)
m 1 m 1
− −
E = f u ui = f(u ) f ui
2 1 0 1 0 − 1
i 1(i[m=1od 2)(cid:8) (cid:0) (cid:1)(cid:9) i 1([im=1od 2)(cid:8)(cid:12) (cid:0) (cid:1)(cid:12)(cid:9)
≡ ≡ (cid:12) (cid:12)
m 1 m 1
− −
= l l = l = l ,l , ,l
0 ni ni 2n 4n n(m 1)
{| − |} { } ··· −
i 1(i[m=1od 2) i 1(i[m=1od 2) (cid:8) (cid:9)
≡ ≡
m 2
−
E = f uiui :1 j n 1
3 1 j j+1 ≤ ≤ −
i 1(i[m=1od 2)(cid:8) (cid:0) (cid:1) (cid:9)
≡
m 2
−
= f ui f ui :1 j n 1
j − j+1 ≤ ≤ −
i 1(i[m=1od 2)(cid:8)(cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12) (cid:9)
≡ (cid:12) (cid:12)
m 2
−
= l l :1 j n 1
n(i 1)+2j 1 n(i 1)+2j+1
− − − − ≤ ≤ −
i 1(i[m=1od 2)(cid:8)(cid:12) (cid:12) (cid:9)
≡ (cid:12) (cid:12)
m 2
−
= l :1 j n 1
n(i 1)+2j
− ≤ ≤ −
i 1(i[m=1od 2)(cid:8) (cid:9)
≡
m 2
−
= l ,l , ,:l
n(i 1)+2 n(i 1)+4 n(i 1)+2(n 1)
− − ··· − −
i 1(i[m=1od 2)(cid:8) (cid:9)
≡
= l ,l ,...,l l ,l , ,l
2 2n+2 n(m 3)+2 4 2n+4 n(m 3)+4
− ∪ ··· − ∪···
(cid:8) l2n 2,l4n 2,...,ln(m(cid:9)3)+(cid:8)2n 2 , (cid:9)
∪ − − − −
(cid:8) (cid:9)
LucasGracefulLabelingforSomeGraphs 5
m 2
−
E = f ui ui :1 j n 1
4 1 j j+1 ≤ ≤ −
i 1([im=1od 2)(cid:8) (cid:0) (cid:1) (cid:9)
≡
m 2
−
= f ui f ui :1 j n 1
j − j+1 ≤ ≤ −
i 1([im=1od 2)(cid:8)(cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12) (cid:9)
≡ (cid:12) (cid:12)
m 2
−
= l l :1 j n 1
ni 2j+2 ni 2j
{| − − − | ≤ ≤ − }
i 1([im=1od 2)
≡
m 2
−
= l :1 j n 1
ni 2j+1
{ − ≤ ≤ − }
i 1([im=1od 2)
≡
m 2
−
= l ,l , ,l
ni 1 ni 3 ni (2n 3)
− − ··· − −
i 1([im=1od 2)(cid:8) (cid:9)
≡
= l ,l , ,l ,l ,l , ,l ,l , ,l .
2n 1 2n 3 3 4n 1 4n 3 2n+3 n(m 1) 1 n(m 1) (2n 3)
− − ··· − − ··· − − ··· − − −
(cid:8) (cid:9)
For n 1(mod 3), let
≡
n−1
3
E = f um um :3s 2 j 3s 1
5 1 j j+1 − ≤ ≤ −
s=1
[ (cid:8) (cid:0) (cid:1) (cid:9)
n−1
3
= f um f um :3s 2 j 3s 1
j − j+1 − ≤ ≤ −
s=1
[ (cid:8)(cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12) (cid:9)
n−1 (cid:12) (cid:12)
3
= l l :3s 2 j 3s 1
n(m 1)+2j 3s+2 n(m 1)+2j 3s+4
− − − − − − ≤ ≤ −
s=1
[ (cid:8)(cid:12) (cid:12) (cid:9)
n−1 (cid:12) (cid:12) n−1
3 3
= l :3s 2 j 3s 1 = l ,l
n(m 1)+2j 3s+2 n(m 1)+3s 1 n(m 1)+3s+1
− − − ≤ ≤ − − − −
s=1 s=1
[ (cid:8) (cid:9) [ (cid:8) (cid:9)
= l ,l ,l ,l , ,l ,l .
n(m 1)+2 n(m 1)+4 n(m 1)+5 n(m 1)+7 n m 2 mn
− − − − ··· −
(cid:8) (cid:9)
We find the edge labeling between the end vertex of sth loop and the starting vertex of
n 1
(s+1)th loop and s=1,2, , − . Let
··· 3
n−1 n−1
3 3
E = f um um = f(um) f um
6 1 3s 3s+1 3s − 3s+1
s=1 s=1
[ (cid:8)(cid:12) (cid:0) (cid:1)(cid:12)(cid:9) [ (cid:8)(cid:12) (cid:0) (cid:1)(cid:12)(cid:9)
= f(u(cid:12)m) f(um) ,(cid:12)f(um) f(cid:12)(um) , f(um) f(cid:12)(um) , , f um f(um)
| 3 − 4 | | 6 − 7 | | 9 − 10 | ··· n−1 − n
= (cid:8) ln(m 1)+5 ln(m 1)+4 , ln(m 1)+8 ln(m 1)+7 , , ln(m(cid:12)1)(cid:0)+n+1 (cid:1) ln(m 1)+(cid:12)n(cid:9)
− − − − − − ··· −(cid:12) − − (cid:12)
= (cid:8)ln(cid:12)(m 1)+3,ln(m 1)+6, (cid:12),(cid:12)ln(m 1)+n 1 = ln(m (cid:12)1)+3,ln(cid:12)(m 1)+6, ,lnm 1 . (cid:12)(cid:9)
(cid:12) − − ···(cid:12) (cid:12) − − −(cid:12) (cid:12) − ··· − (cid:12)
(cid:8) (cid:9) (cid:8) (cid:9)