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International Journal of Mathematical Combinatorics, Vol. 1, 2011 ISSN 1937 - 1055 VOLUME 1, 2011 INTERNATIONAL JOURNAL OF MATHEMATICAL COMBINATORICS EDITED BY THE MADIS OF CHINESE ACADEMY OF SCIENCES March, 2011 Vol.1, 2011 ISSN 1937-1055 International Journal of Mathematical Combinatorics Edited By The Madis of Chinese Academy of Sciences March, 2011 Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sci- ences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects 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 gravitational fields; 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 also welcome. It is also available from the below international databases: Serials Group/Editorial Department of EBSCO Publishing 10 Estes St. Ipswich, MA 01938-2106, USA Tel.: (978) 356-6500, Ext. 2262 Fax: (978) 356-9371 http://www.ebsco.com/home/printsubs/priceproj.asp and Gale Directory of Publications and Broadcast Media, Gale, a part of Cengage Learning 27500 Drake Rd. Farmington Hills, MI 48331-3535, USA Tel.: (248) 699-4253, ext. 1326; 1-800-347-GALE Fax: (248) 699-8075 http://www.gale.com Indexing and Reviews: Mathematical Reviews(USA), Zentralblatt fur Mathematik(Germany), Referativnyi Zhurnal (Russia), Mathematika (Russia), Computing Review (USA), Institute for Scientific Information (PA, USA), Library of Congress Subject Headings (USA). Subscription A subscription can be ordered by a mail or an email directly to Linfan Mao The Editor-in-Chief of International Journal of Mathematical Combinatorics Chinese Academy of Mathematics and System Science Beijing, 100190, P.R.China Email: [email protected] Price: US$48.00 Editorial Board H.Iseri Mansfield University, USA Email: hiseri@mnsfld.edu Editor-in-Chief Linfan MAO M.Khoshnevisan Chinese Academy of Mathematics and System School of Accounting and Finance, Griffith University, Australia Science, P.R.China Email: [email protected] Xueliang Li Nankai University, P.R.China Email: [email protected] Editors Han Ren East China Normal University, P.R.China S.Bhattacharya Email: [email protected] Deakin University Geelong Campus at Waurn Ponds W.B.Vasantha Kandasamy Australia Indian Institute of Technology, India Email: [email protected] 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 Academy of Mathematics and System 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 Academy of Mathematics and System Science, P.R.China Email: [email protected] Yuanqiu Huang Hunan Normal University, P.R.China Email: [email protected] ii International Journal of Mathematical Combinatorics Achievement provides the only real pleasure in life. By Thomas Edison, an American inventor. International J.Math. Combin. Vol.1 (2011), 01-19 Lucas Graceful Labeling for Some Graphs M.A.Perumal1, S.Navaneethakrishnan2 and A.Nagarajan2 1. Department of Mathematics, National Engineering College, K.R.Nagar, Kovilpatti, Tamil Nadu, India 2. Department of Mathematics, V.O.C College, Thoothukudi, Tamil Nadu, 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 ǫ N) is said to be Lucas graceful labeling if the induced → { · · · } 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 is said to be strong Lucas graceful labeling if the induced edge → { · · · } 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, P e, Sm,n, Fm@Pn, Cm@Pn, K1,n 2Pm, C3@2Pn and n − ⊙ Cn@K1,2 admit Lucas graceful labeling and some graphs namely K1,n and Fn admit strong Lucas graceful labeling. Key Words: Smarandache-Fibonacci triple, super Smarandache-Fibonacci graceful graph, 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 Pn. A cycle of length n is denoted by Cn.G is a graph obtained from the graph G by attaching a pendant vertex to each vertex of G. The concept of graceful labeling 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 1Received November 11, 2010. Accepted February 10, 2011. 2 M.A.Perumal, S.Navaneethakrishnan and A.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 , (aǫN), → { 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, ,. Then G is called Lucas graceful graph if it admits Lucas graceful labeling. Also 5 · · · an injective function f : V (G) l ,l ,l , ,l is said to be strong Lucas graceful labeling if →{ 0 1 2 · · · q} the induced edge labeling f (uv)= f(u) f(v) is a bijection onto the set l ,l , ,l . Then 1 | − | { 1 2 · · · q} G is called strong Lucas graceful graph if it admits strong Lucas graceful labeling. In this paper, we show that some graphs namely P , P + e, S , F @P , C @P , K 2P , C @2P n n − m,n m n m n 1,n ⊙ m 3 n and Cn@K1,2 admit Lucas graceful labeling and some graphs namely K1,n and Fn admit strong 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 } (aǫN) is said to be 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, ,.
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