ISSN 1937 - 1055 VOLUME 1, INTERNATIONAL MATHEMATICAL JOURNAL 2015 OF COMBINATORICS EDITED BY THE MADIS OF CHINESE ACADEMY OF SCIENCES AND ACADEMY OF MATHEMATICAL COMBINATORICS & APPLICATIONS March, 2015 Vol.1, 2015 ISSN 1937-1055 International Journal of Mathematical Combinatorics Edited By The Madis of Chinese Academy of Sciences and Academy of Mathematical Combinatorics & Applications March, 2015 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 Sciences 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; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Differential Geometry; Geometry on manifolds; 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 Combinatorics 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, 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Academy of Mathematics and System Science, P.R.China and Academy of Mathematical Combinatorics & Applications, USA Email: maolinfan@163.com Deputy Editor-in-Chief Shaofei Du Capital Normal University, P.R.China Email: dushf@mail.cnu.edu.cn Baizhou He Beijing University of Civil Engineering and Architecture, P.R.China Email: hebaizhou@bucea.edu.cn Xiaodong Hu Chinese Academy of Mathematics and System Science, P.R.China Email: xdhu@amss.ac.cn Guohua Song Beijing University of Civil Engineering and Yuanqiu Huang Hunan Normal University, P.R.China Architecture, P.R.China Email: hyqq@public.cs.hn.cn Email: songguohua@bucea.edu.cn Editors H.Iseri Mansfield University, USA Email: hiseri@mnsfld.edu S.Bhattacharya Xueliang Li Deakin University Nankai University, P.R.China Geelong Campus at Waurn Ponds Email: lxl@nankai.edu.cn Australia Email: Sukanto.Bhattacharya@Deakin.edu.au Guodong Liu Huizhou University Said Broumi Email: lgd@hzu.edu.cn Hassan II University Mohammedia W.B.Vasantha Kandasamy Hay El Baraka Ben M’sik Casablanca Indian Institute of Technology, India B.P.7951 Morocco Email: vasantha@iitm.ac.in Junliang Cai Ion Patrascu Beijing Normal University, P.R.China Fratii Buzesti National College Email: caijunliang@bnu.edu.cn Craiova Romania Yanxun Chang Han Ren Beijing Jiaotong University, P.R.China East China Normal University, P.R.China Email: yxchang@center.njtu.edu.cn Email: hren@math.ecnu.edu.cn Jingan Cui Beijing University of Civil Engineering and Ovidiu-Ilie Sandru Politechnica University of Bucharest Architecture, P.R.China Romania Email: cuijingan@bucea.edu.cn ii International Journal of Mathematical Combinatorics Mingyao Xu Peking University, P.R.China Email: xumy@math.pku.edu.cn Guiying Yan Chinese Academy of Mathematics and System Science, P.R.China Email: yanguiying@yahoo.com Y Zhang Department of Computer Science Georgia State University, Atlanta, USA Famous Words: Nothing in life is to be feared It is only to be understood By Marie Curie, a Polish and naturalized-French physicist and chemist International J.Math Combin Vol.1(2015), 1-13 N ∗ C ∗ − Smarandache Curves of Mannheim Curve Couple According to Frenet Frame Să uleyman S ENYURT and Abdussamet C ALISáKAN (Faculty of Arts and Sciences, Department of Mathematics, Ordu University, 52100, Ordu/Turkey) E-mail: senyurtsuleyman@hotmail.com Abstract: In this paper, when the unit Darboux vector of the partner curve of Mannheim curve are taken as the position vectors, the curvature and the torsion of Smarandache curve are calculated These values are expressed depending upon the Mannheim curve Besides, we illustrate example of our main results Key Words: Mannheim curve, Mannheim partner curve, Smarandache Curves, Frenet invariants AMS(2010): 53A04 §1 Introduction A regular curve in Minkowski space-time, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarandache curve ([12]) Special Smarandache curves have been studied by some authors Melih Turgut and Să uha Ylmaz studied a special case of such curves and called it Smarandache T B2 curves in the space E14 ([12]) Ahmad T.Ali studied some special Smarandache curves in the Euclidean space He studied Frenet-Serret invariants of a special case ([1]) Muhammed C ¸ etin , Yılmaz Tun¸cer and Kemal Karacan investigated special Smarandache curves according to Bishop frame in Euclidean 3-Space and they gave some differential goematric properties of Smarandache curves, also they found the centers of the osculating spheres and curvature spheres of Smarandache curves ([5]) S¸enyurt and C ¸ alı¸skan investigated special Smarandache curves in terms of Sabban frame of spherical indicatrix curves and they gave some characterization of ă Smarandache curves ([4]) Ozcan Bektaás and Salim Yă uce studied some special Smarandache ă curves according to Darboux Frame in E ([2]) Nurten Bayrak, Ozcan Bektaás and Salim Yă uce studied some special Smarandache curves in E1 [3] Kemal Tas.kăopră u, Murat Tosun studied special Smarandache curves according to Sabban frame on S ([11]) In this paper, special Smarandache curve belonging to α∗ Mannheim partner curve such as N ∗ C ∗ drawn by Frenet frame are defined and some related results are given Received September 8, 2014, Accepted February 12, 2015 Să uleyman S ¸ ENYURT and Abdussamet C ¸ ALIS ¸ KAN §2 Preliminaries The Euclidean 3-space E be inner product given by , = x21 + x32 + x23 where (x1 , x2 , x3 ) ∈ E Let α : I → E be a unit speed curve denote by {T, N, B} the moving Frenet frame For an arbitrary curve α ∈ E , with first and second curvature, κ and τ respectively, the Frenet formulae is given by ([6], [9])  ′  T = κN     N ′ = −κT + τ B (2.1) B ′ = −τ N For any unit speed α : I → E3 , the vector W is called Darboux vector defined by W = τ (s)T (s) + κ(s) + B(s) If consider the normalization of the Darboux C = cos ϕ = W , we have W τ (s) κ(s) , sin ϕ = , W W C = sin ϕT (s) + cos ϕB(s) (2.2) where ∠(W, B) = ϕ Let α : I → E3 and α∗ : I → E3 be the C − class differentiable unit speed two curves and let {T (s), N (s), B(s)} and {T ∗ (s), N ∗ (s), B ∗ (s)} be the Frenet frames of the curves α and α∗ , respectively If the principal normal vector N of the curve α is linearly dependent on the binormal vector B of the curve α∗ , then (α) is called a Mannheim curve and (α∗ ) a Mannheim partner curve of (α) The pair (α, α∗ ) is said to be Mannheim pair ([7], [8]) The relations between the Frenet frames {T (s), N (s), B(s)} and {T ∗ (s), N ∗ (s), B ∗ (s)} are as follows:  ∗   T = cos θT − sin θB    where ∠(T, T ∗ ) = θ ([8]) N ∗ = sin θT + cos θB (2.3) B∗ = N   cos θ = ds∗ ds ds∗  sin θ = λτ ∗ ds (2.4) Theorem 2.1([7]) The distance between corresponding points of the Mannheim partner curves in E3 is constant N ∗ C ∗ − Smarandache Curves of Mannheim Curve Couple According to Frenet Frame Theorem 2.2 Let (α, α∗ ) be a Mannheim pair curves in E3 For the curvatures and the torsions of the Mannheim curve pair (α, α∗ ) we have,  ds∗ ∗   κ = τ sin θ ds  (2.5)    τ = −τ ∗ cos θ ds∗ ds and  dθ κ   = θ′ √ κ∗ =   ds∗  λτ κ2 + τ (2.6)     τ ∗ = (κ sin θ − τ cos θ) ds∗ ds Theorem 2.3 Let (α, α∗ ) be a Mannheim pair curves in E3 For the torsions τ ∗ of the Mannheim partner curve α∗ we have κ τ∗ = λτ Theorem 2.4([10]) Let (α, α∗ ) be a Mannheim pair curves in E3 For the vector C ∗ is the direction of the Mannheim partner curve α∗ we have C∗ = 1+ θ′ W C+ θ′ W 1+ θ′ W N (2.7) where the vector C is the direction of the Darboux vector W of the Mannheim curve α §3 N ∗ C ∗ − Smarandache Curves of Mannheim Curve Couple According to Frenet Frame Let (α, α∗ ) be a Mannheim pair curves in E and {T ∗ N ∗ B ∗ } be the Frenet frame of the Mannheim partner curve α∗ at α∗ (s) In this case, N ∗ C ∗ - Smarandache curve can be defined by β1 (s) = √ (N ∗ + C ∗ ) (3.1) Solving the above equation by substitution of N ∗ and C ∗ from (2.3) and (2.7), we obtain β1 (s) = cos θ W + sin θ θ′ + W T + θ′ N + cos θ θ′ + W θ′ + W − sin θ W B (3.2) Să uleyman S ENYURT and Abdussamet C ALIS ¸ KAN The derivative of this equation with respect to s is as follows, θ ′2 + Tβ1 (s) = ′ W √ W ′ W √ θ ′2 + W √ θ ′ κ cos θ λτ W cos θ − 2 θ ′2 + W θ′ + ′ W √ √ θ ′2 + W + θ ′2 + ) κ λτ W θ ′2 + ′2 κ(θ + W λτ W ) W θ′ W N ′ W −2 √ θ ′2 + W ′ W ′ W − √ W − √ 2 θ ′2 + W θ′ κ(θ ′2 + W λτ W θ ′ κ sin θ λτ W + κ λτ T+ θ′ sin θ B κ λτ W −2 √ ′ W θ ′2 + W · θ′ (3.3) In order to determine the first curvature and the principal normal of the curve β1 (s), we formalize √ (r¯1 cos θ + r¯2 sin θ)T + r¯3 N + (−r¯1 sin θ + r¯2 cos θ)B Tβ′ (s) = √ ′ W θ ′2 + W √ θ ′2 + W θ′ 2 + κ(θ ′2 + W λτ W ) κ λτ W ′ W −2 √ θ ′2 + W θ′ where r¯1 = κ λτ − W θ′ + W θ′ κ λτ W θ′ + W θ′ + W 2 ′ θ′ θ′ + W θ′ − κ λτ ′ θ′ + W W θ′ + W θ′ + W W − κ λτ ′ 2 θ′ κ λτ W θ′ + W W θ′ + W ′ θ′ + W − κ λτ ′ ′ W − ′ θ′ θ′ θ′ W θ′ + W W θ′ + W θ′ θ′ + W θ′ θ′ + W θ′ + W ′ θ′ + W ′ θ′ W θ′ θ′ + W ′ W θ′ + W ′ N ∗ C ∗ − Smarandache Curves of Mannheim Curve Couple According to Frenet Frame θ′ + W × 2 θ′ θ′ + W θ W θ′ θ′ 2 + W W ′2 θ + W θ′ κ λτ W −2κ∗ W θ′ + W κ λτ θ + W = ′ κ λτ ′ W ′ θ′ + W θ′ + W θ′ + W θ′ 2 θ′ θ′ + W ′ θ′ + W ′ ′ θ′ ′ κ λτ θ′ + W ′ θ′ θ′ κ λτ W W , ′2 θ + W θ′ θ′ θ′ + W θ′ θ′ + W θ′ ′ W ′ W ′ θ′ + W W θ′ + W ′ θ′ κ λτ W ′ W 2 ′ θ′ + W κ λτ ′2 θ′ + W θ′ κ λτ W θ′ + W θ′ κ − λτ W W ′ θ′ ′ 2 θ′ + W θ + W W + θ + W −2 κ λτ − τ∗ θ′ + W W − θ′ κ λτ W θ′ κ λτ W − θ′ θ′ + W W θ′ θ′ + W W θ′ κ + λτ W W 2 ′ θ′ θ′ + W θ′ + W θ′ κ λτ W W ′2 θ′ + W θ′ + W θ′ ′ θ′ θ′ θ′ r¯2 2 θ′ + W θ′ 2 θ′ θ′ θ′ + W ′ θ + W θ′ + W ′ θ′ + W ′2 ′ θ′ + W + W θ′ ′ W θ′ W θ′ + W ′ θ′ + W ′2 θ′ + W W κ + λτ W θ′ + W θ′ + W θ′ + W ′ W κ λτ +2 + W ′ ′ θ′ κ − λτ W W κ λτ θ′ κ +2 λτ W W θ′ θ′ + W θ′ + W κ λτ +3 +3 θ′ κ λτ W θ′ κ λτ W Some Characterizations for the Involute Curves in Dual Space 125 On the other hand, from the equation (3.16), we can write − τ − κ ′ = ′ κτ −κ τ λκ(κ2 +τ ) (κ2 +τ ) = τ κ ′ κ2 κ2 +τ · (3.30) λκ Substituting by the equation (3.29) into the equation (3.30), then we find − τ = 0, which completes the proof ¾ References [1] Bilici M and C ¸ alı¸skan M., Some characterizations for the pair of involute-evolute curves in Euclidean space, Bulletin of Pure and Applied Sciences, Vol.21E, No.2, 289-294, 2002 [2] Bilici M and C ¸ alı¸skan M., On the involutes of the spacelike curve with a timelike binormal in Minkowski 3-space, International Mathematical Forum, Vol 4, No.31, 1497-1509, 2009 [3] Bilici M and C ¸ alı¸skan M., Some new notes on the involutes of the timelike curves in Minkowski3-space, Int.J.Contemp.Math Sciences, Vol.6, No.41, 2019-2030, 2011 [4] Bă ukácu ă B and Karacan M.K., On the involute and evolute curves of the spacelike curve with a spacelike binormal in Minkowski space, Int J Contemp Math Sciences, Vol 2, No 5, 221 - 232, 2007 [5] Fenchel W., On the differential geometry of closed space curves, Bull Amer Math Soc., Vol.57, No.1, 44-54, 1951 [6] Hacısaliho˘glu H H., Acceleration Axes in Spatial Kinematics I, Communications, S´erie A: Math´ematiques, Physique et Astronomie, Tome 20 A, pp 1-15, Ann´ee 1971 [7] Hacısaliho˘glu H.H., Differential Geometry, (Turkish) Ankara University of Faculty of Science, 2000 [8] Millman R.S and Parker G.D., Elements of Differential Geometry, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1977 [9] Sabuncuo˘glu, A., Differential Geometry (Turkish), Nobel Publishing, 2006 [10] Sáenyurt S and Gă ur S., On the dual spacelike-spacelike involute-evolute curve couple on dual Lorentzian space, International Journal of Mathematical Engineering and Science, ıssn:2277-6982, vol.1, Issue :5, 14-29, 2012 [11] Sáenyurt S and Gă ur S., Timelike - spacelike involute - evolute curve couple on dual Lorentzian space, J Math Comput Sci., Vol.2, No 6, 1808-1823, 2012 [12] Sáenyurt S and Gă ur S., Spacelike - timelike involute- evolute curve couple on dual Lorentzian space, J Math Comput Sci., Vol.3, No.4,1054-1075, 2013 ă Special involute-evolute partner D-curves in E , European Journal [13] Yă uce S and Bekta¸s O., of Pure and Applied Mathematics, Vol 6, No 1, 20-29, 2013 International J.Math Combin Vol.1(2015), 126-135 One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs V.Ramachandran (Department of Mathematics, P.S.R Engineering College, Sivakasi, Tamil Nadu, India) C.Sekar (Department of Mathematics, Aditanar College of Arts and Science, Tiruchendur, Tamil Nadu, India) E-mail: me.ram111@gmail.com; sekar.acas@gmail.com Abstract: A graph G is said to be one modulo N graceful (where N is a positive integer) if there is a function φ from the vertex set of G to {0, 1, N, (N + 1), 2N, (2N + 1), · · · , N (q − 1), N (q − 1) + 1}in such a way that (i) φ is − (ii) φ induces a bijection φ∗ from the edge set of G to {1, N + 1, 2N + 1, · · · , N (q − 1) + 1}where φ∗ (uv)=|φ(u) − φ(v)| In this paper we prove that arbitrary supersubdivision of disconnected path and cycle Pn ∪ Cr is one modulo (1) N graceful for all positive integer N Also we prove that the graph Pn+ − vk is one modulo N graceful for every positive integer N Key Words: Graceful, modulo N graceful, disconnected graphs, arbitrary supersubdivision graphs, Pn (1) Cn and Pn+ − vk AMS(2010): 05C78 §1 Introduction S W Golomb [3] introduced graceful labelling Odd gracefulness was introduced by R B Gnanajothi [4] C Sekar [11] introduced one modulo three graceful labelling In [8,9], we introduced the concept of one modulo N graceful where N is any positive integer In the case N = 2, the labelling is odd graceful and in the case N = the labelling is graceful Joseph A Gallian [2] surveyed numerous graph labelling methods Recently G Sethuraman and P Selvaraju [5] have introduced a new method of construction called supersubdivision of a graph Let G be a graph with n vertices and t edges A graph H is said to be a supersubdivision of G if H is obtained by replacing every edge ei of G by the complete bipartite graph K2,m for some positive integer m in such a way that the ends of ei are merged with the two vertices part of K2,m after removing the edge ei from G A supersubdivision H of a graph G is said to be an arbitrary supersubdivision of the graph G if every edge of G is replaced by an arbitrary K2,m (m may vary for each edge arbitrarily) A graph G is said to be connected if any two vertices of G are joined by a path Otherwise it is called disconnected graph G Sethuraman and P Selvaraju [6] proved that every connected graph has some supersub1 Received July 15, 2014, Accepted March 12, 2015 127 One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs division that is graceful They pose the question as to whether some supersubdivision is valid for disconnected graphs [10] We proved that an arbitrary supersubdivision of disconnected paths are graceful Barrientos and Barrientos [1] proved that any disconnected graph has a supersubdivision that admits an α-labeling They also proved that every supersubdivision of a connected graph admits an α-labeling In this paper we prove that arbitrary supersubdivision of disconnected path and cycle Pn ∪ Cr is one modulo N graceful for all positive integer N When N = we get an affirmative (1) answer for their question Also we prove that the graph Pn+ − vk is one modulo N graceful for every positive integer N §2 Main Results Definition 2.1 A graph G with q edges is said to be one modulo N graceful (where N is a positive integer) if there is a function φ from the vertex set of G to {0, 1, N, (N + 1), 2N, (2N + 1), , N (q − 1), N (q − 1) + 1} in such a way that (i) φ is − (ii) φ induces a bijection φ∗ from the edge set of G to {1, N + 1, 2N + 1, , N (q − 1) + 1}where φ∗ (uv)=|φ(u) − φ(v)| Definition 2.2 In the complete bipartite graph K2,m we call the part consisting of two vertices, the 2-vertices part of K2,m and the part consisting of m vertices the m-vertices part of K2,m Let G be a graph with p vertices and q edges A graph H is said to be a supersubdivision of G if H is obtained by replacing every edge e of G by the complete bipartite graph K2,m for some positive integer m in such a way that the ends of e are merged with the two vertices part of K2,m after removing the edge e from G H is denoted by SS(G) Definition 2.3 A supersubdivision H of a graph G is said to be an arbitrary supersubdivision of the graph G if every edge of G is replaced by an arbitrary K2,m (m may vary for each edge arbitrarily) H is denoted by ASS(G) (1) (1) (1) Definition 2.4 Let v1 , v2 , , be the vertices of a path of length n and v1 , v2 , , be the pendant vertices attached with v1 , v2 , , respectively The removal of a pendant vertex (1) (1) vk where ≤ k ≤ n from Pn+ yields the graph Pn+ − vk Theorem 2.5 Arbitrary supersubdivision of disconnected path and cycle Pn ∪ Cr is one modulo N graceful provided the arbitrary supersubdivision is obtained by replacing each edge of G by K2,m with m Proof Let Pn be a path with successive vertices v1 , v2 , · · · , and let ei (1 ≤ i ≤ n − 1) denote the edge vi vi+1 of Pn Let Cr be a cycle with successive vertices vn+1 , vn+2 , · · · , vn+r and let ei (n + ≤ i ≤ n + r) denote the edge vi vi+1 Let H be an arbitrary supersubdivision of the disconnected graph Pn ∪ Cr where each edge ei of Pn ∪ Cr is replaced by a complete bipartite graph K2,mi with mi for ≤ i ≤ n − and n + ≤ i ≤ n + r Here the edge vn+r vn+1 is replaced by k2,r−1 We observe that H has M = 2(m1 + m2 + · · · + mn−1 + mn+1 + · · · + mn+r ) edges 128 V.Ramachandran and C.Sekar v6 (2) (1) v23 (1) v12 (2) (1) (m ) (4) (m ) v64 = v64 (2) v23 (m ) v12 = v12 v2 (3) (m ) v23 = v23 v5 v45 = v45 v4 v1 (m ) v56 = v56 v (1) 56 (1) v64 (3) v45 v3 (2) v45 (1) v45 Figure Supersubdivision of P3 ∪ C3 Define φ(vi ) = N (i − 1), i = 1, 2, 3, · · · , n, φ(vi ) = N (i), i = n + 1, n + 2, n + 3, · · · , n + r, and for k = 1, 2, 3, , mi ,   N (M − 2k + 1) + if i =         N (M − + i) + − 2N (m1 + m2 + · · · + mi−1 + k − 1) if i = 2, 3, n − (k) φ(vi,i+1 ) = N (M − + i) + − 2N (m1 + m2 + · · · + mn−1 + k − 1) if i = n +     N (M − + i) + − 2N [(m1 + m2 + · · · + mn−1 )+      (mn+1 + · · · + mi−1 ) + k − 1] if i = n + 2, n + 3, n + r − (k) and for k = 1, 2, 3, · · · , mn+r , φ(vn+r,n+1 ) = N (n + r − k + mn+r ) + From the definition of φ it is clear that {φ(vi ), i = 1, 2, · · · , n + r} (k) {φ(vi,i+1 ), i = 1, 2, · · · , n + r − and k = 1, 2, 3, · · · , mi } = {0, N, 2N, · · · , N (n − 1)} (k) {φ(vn+r,n+1 ), k = 1, 2, 3, · · · , mi } {N (n + 1), N (n + 2), · · · , N (n + r)} {N [M − 2k + 1] + 1, N [M − 2m1 ] + 1, N [M − 2m1 − 2] + · · · , N [M − 2(m1 + m2 ) + 2] + 1, N [M − 2(m1 + m2 ) + 1] + 1, N [M − 2(m1 + m2 ) − 1] + 1, , N [M − 2(m1 + m2 + m3 ) + 3] + 1, , N [M − + n − 2(m1 + m2 + · · · + mn−2 )] + 1, N [M − + n − 2(m1 + m2 + · · · + mn−2 )] + 1, , N [M − + n − 2(m1 + m2 + · · · + mn−1 )] + 1, N [M + n − 2(m1 + m2 + · · · + mn−1 )] + 1, N [M + n − 2(m1 + m2 + · · · + mn−1 + 1)] + 1, , N [M + n − 2(m1 + m2 + · · · + mn−1 + mn+1 − 1)] + 1, N [M + + n − 2(m1 + m2 + · · · + mn−1 + mn+1 )] + 1, 129 One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs N [M − + n − 2(m1 + m2 + · · · + mn−1 + mn+1 )] + 1, · · · , N [M + + n − 2(m1 + m2 + · · · + mn−1 + mn+1 + mn+2 )] + 1, N [M + + n − 2(m1 + m2 + · · · + mn−1 + mn+1 + mn+2 )] + 1, N [M + n − 2(m1 + m2 + · · · + mn−1 + mn+1 + mn+2 )] + 1, , N [M + + n − 2(m1 + m2 + · · · + mn−1 + mn+1 + mn+2 + mn+3 )] + 1, N [M − + n + r − 2(m1 + m2 + · · · + mn−1 + mn+1 + mn+2 )] + 1, N [M − + n + r − 2[(m1 + m2 + · · · + mn−1 ) + (mn+1 + mn+2 + · · · + mn+r−2 )]] + 1, N [M − + n + r − 2[(m1 + m2 + · · · + mn−1 ) + (mn+1 + mn+2 + · · · + mn+r−2 )]] + 1, · · · , N [M + n + r − 2[(m1 + m2 + · · · + mn−1 ) + (mn+1 + mn+2 + · · · + mn+r−1 )]] + 1} {N (n + r − + mn+r ) + 1, N (n + r − + mn+r ) + 1, · · · , N (n + r) + 1} Thus it is clear that the vertices have distinct labels Therefore φ is − We compute the edge labels as follows: (k) (k) (k) For k = 1, 2, · · · , m1 , φ∗ (v1,2 v1 ) =| φ(v1,2 ) − φ(v1 ) | = N (M − 2k + 1) + 1, φ∗ (v1,2 v2 ) =| (k) φ(v1,2 ) − φ(v2 ) | = N (M − 2k) + (k) (k) For k = 1, 2, · · · , mi and i = 2, 3, · · · , n − 1, φ∗ (vi,i+1 vi ) =| φ(vi,i+1 ) − φ(vi ) | = N (M − (k) (k) 2k + 1) − 2N (m1 + m2 + · · · + mi−1 ) + 1, φ∗ (vi,i+1 vi+1 ) =| φ(vi,i+1 ) − φ(vi+1 ) | = N (M − 2k) − 2N (m1 + m2 + · · · + mi−1 ) + (k) (k) For k = 1, 2, · · · , mn+1 , φ∗ (vn+1,n+2 vn+1 ) =| φ(vn+1,n+2 ) − φ(vn+1 ) | = N (M − 2k + 1) − (k) (k) 2N (m1 + m2 + · · · + mn−1 ) + 1, φ∗ (vn+1,n+2 vn+2 ) =| φ(vn+1,n+2 ) − φ(vn+2 ) | = N (M − 2k) − 2N (m1 + m2 + · · · + mn−1 ) + (k) (k) For k = 1, 2, , mi and j = n + 2, n + 3, · · · , n + r, φ∗ (vi,i+1 vi ) =| φ(vi,i+1 ) − φ(vi ) | = (k) N (M −2k+1)−2N {(m1 +m2 +· · ·+mn−1 )+(mn+1 +mn+2 +· · ·+mi−1 )}+1, φ∗ (vi,i+1 vi+1 ) =| (k) φ(vi,i+1 )−φ(vi+1 ) | = N (M −2k)−2N {(m1 +m2 +· · ·+mn−1 )+(mn+1 +mn+2 +· · ·+mi−1 )}+1 (k) (k) For k = 1, 2, · · · , mn+r , φ∗ (vn+r,n+1 vn+r ) =| φ(vn+r,n+1 ) − φ(vn+r ) | = N (mn+r − k) + 1, (k) (k) φ∗ (vn+r,n+1 vn+1 ) =| φ(vn+r,n+1 ) − φ(vn+1 ) | = N (mn+r + r − k − 1) + It is clear from the above labelling that the mi +2 vertices of K2,mi have distinct labels and the 2mi edges of K2,mi also have distinct labels for ≤ i ≤ n − and n + ≤ i ≤ n + r − Therefore the vertices of each K2,mi , ≤ i ≤ n − and n + ≤ i ≤ n + r − in the arbitrary supersubdivision H of Pn ∪Cr have distinct labels and also the edges of each K2,mi , ≤ i ≤ n−1 and n + ≤ i ≤ n + r − in the arbitrary supersubdivision graph H of Pn ∪ Cr have distinct labels Clearly H is one modulo N graceful Hence arbitrary supersubdivisions of disconnected path and cycle Pn ∪ Cr is one modulo N graceful, for every positive integer N Consequently, every disconnected graph has some supersubdivision that is one modulo N ¾ graceful Example 2.6 A odd graceful labelling of ASS(P3 ∪ C4 ) is shown in Figure 130 V.Ramachandran and C.Sekar 12 27 14 29 33 53 19 59 17 15 49 10 35 55 39 45 43 47 Figure Example 2.7 A graceful labelling of ASS(P3 ∪ C3 ) is shown in Figure 11 23 26 21 14 24 13 19 16 18 20 Figure (1) Theorem 2.8 For any pendant vertex vk graceful for every positive integer N (1) ∈ V (Pn+ ), the graph Pn+ − vk is one modulo N 131 One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs (1) (1) (1) Proof Let v1 , v2 , · · · , be the vertices of a path of length n and v1 , v2 , · · · , the (1) pendant vertices attached with v1 , v2 , · · · , respectively Consider the graph Pn+ − vk , where ≤ k ≤ n It has 2n − vertices and 2n − edges Case n is even and k is even Define   N (2n − 3) + − 2N (i − 1) for i = 1, 2, · · · , k φ(v2i−1 ) =  N (2n − 3) + − 2N ( k − 1) − N − 2N (i − ( k + 1)) for i = 2 k + 1, · · · , n2 , φ(v2i ) = N (2i − 1) for i = 1, 2, · · · , n2 ,   2N (n − 2) + − 2N (i − 1) for i = 1, 2, · · · , k − (1) φ(v2i ) =  2N (n − 2) + − 2N ( k − 2) − 3N − 2N (i − ( k + 1)) fori = 2 k + 1, k2 + 2, · · · , n2 , (1) φ(v2i−1 ) = 2N (i − 1) for i = 1, 2, · · · , n2 From the definition of φ it is clear that n n {φ(v2i−1 ), i = 1, 2, · · · , } {φ(v2i ), i = 1, 2, · · · , } 2 k k k n (1) {φ(v2i ), i = 1, 2, · · · , − 1, + 1, + 2, · · · , } 2 2 n (1) {φ(v2i−1 ), i = 1, 2, · · · , } = {N (2n − 3) + 1, N (2n − 5) + 1, · · · , N (2n − k − 1) + 1, N (2n − k − 2) + 1, N (2n − k − 4) + 1, , N n + 1} {N, 3N, · · · , N (n − 1)} {2N (n − 2) + 1, 2N (n − 3) + 1, · · · , N (2n − k) + 1, N (2n − k − 3) + 1, N (2n − k − 5) + 1, · · · , N (n − 1) + 1} {0, 2N, , N (n − 2)} Thus it is clear that the vertices have distinct labels Therefore φ is − We compute the edge labels as follows (1) For i = 1, 2, · · · , k2 , φ∗ (v2i−1 v2i )=| φ(v2i−1 ) − φ(v2i ) |= N (2n − 4i) + 1, φ∗ (v2i−1 v2i−1 ) = (1) | φ(v2i−1 ) − φ(v2i−1 ) |= N (2n − 4i + 1) + (1) For i = 1, 2, · · · , k2 −1, φ∗ (v2i+1 v2i ) =| φ(v2i+1 )−φ(v2i ) |= N (2n−4i−2)+1, φ∗ (v2i v2i ) = (1) | φ(v2i ) − φ(v2i ) |= N (2n − 4i − 1) + For i = k2 + 1, k2 + 2, · · · , n2 , φ∗ (v2i−1 v2i )=| φ(v2i−1 ) − φ(v2i ) |= N (2n − 4i + 1) + 1, (1) (1) (1) (1) φ∗ (v2i−1 v2i−1 )=| φ(v2i−1 ) − φ(v2i−1 ) |= N (2n − 4i + 2) + 1, φ∗ (v2i v2i )=| φ(v2i ) − φ(v2i ) |= N (2n − 4i) + For i = k + 1, k2 + 2, · · · , n2 − 1, φ∗ (v2i+1 v2i )=| φ(v2i+1 ) − φ(v2i ) | = N (2n − 4i − 1) + This show that the edges have the distinct labels {1, N + 1, 2N + 1, · · · , N (q − 1) + 1}, (1) where q = 2n − Hence for every positive integer N , Pn+ − vk is one modulo N graceful if n is even and k is even 132 V.Ramachandran and C.Sekar (1) + Example 2.9 A one modulo 10 graceful labelling of P10 − v6 171 10 151 30 161 20 141 131 50 is shown in Figure 121 70 101 90 60 111 80 91 40 Figure Case n is even and k is odd Define   N (2i − 1) for i = 1, 2, · · · , k−1 φ(v2i ) =  N (k − 2) + N + 2N (i − ( (k+1) )) f or i = k+1 k+3 , ,··· , , n2 φ(v2i−1 ) = N (2n − 3) + − 2N (i − 1) for i = 1, 2, · · · , n2 , (1) φ(v2i−1 ) (1)   2N (i − 1) for i = 1, 2, · · · , k−1 =  2N ( k−1 − 1) + 3N + 2N (i − ( k+3 )) fori = 2 k+3 k+5 , ,··· , n2 , φ(v2i ) = 2N (n − 2) + − 2N (i − 1) for i = 1, 2, · · · , n2 (1) The proof is similar to that of Case Hence for every positive integer N , Pn+ − vk is one modulo N graceful if n is even and k is odd (1) + Example 2.10 A one modulo graceful labelling of P12 − v9 85 77 12 69 20 61 28 81 73 16 65 24 57 Figure Case n is odd and k is even Define 53 is shown in Figure 32 45 40 49 36 41 133 One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs   N (2n − 3) + − 2N (i − 1) for i = 1, 2, · · · , k φ(v2i−1 ) =  N (2n − 3) + − 2N ( k − 1) − N − 2N (i − ( k + 1)) fori = 2 k + 1, · · · , n−1   2N (n − 2) + − 2N (i − 1) for i = 1, 2, , k − (1) φ(v2i ) =  2N (n − 2) + − 2N ( k − 2) − 3N − 2N (i − ( k + 1)) fori = 2 k + 1, · · · , n−1 , φ(v2i ) = N (2i − 1) for i = 1, 2, , n−1 , , (1) φ(v2i−1 ) = 2N (i − 1) for i = 1, 2, , n−1 From the definition of φ it is clear that n−1 n−1 } {φ(v2i ), i = 1, 2, · · · , } 2 k k k n−1 n−1 (1) (1) {φ(v2i ), i = 1, 2, · · · , − 1, + 1, + 2, · · · , } {φ(v2i−1 ), i = 1, 2, · · · , } 2 2 = {N (2n − 3) + 1, N (2n − 5) + 1, , N (2n − k − 1) + 1, N (2n − k − 2) + 1, {φ(v2i−1 ), i = 1, 2, · · · , N (2n − k − 4) + 1, , N (n − 1) + 1} {N, 3N, , N (n − 2)} {2N (n − 2) + 1, 2N (n − 3) + 1, , N (2n − k) + 1, N (2n − k − 3) + 1, N (2n − k − 5) + 1, , N n + 1} {0, 2N, , N (n − 1)} Thus it is clear that the vertices have distinct labels Therefore φ is − We compute the edge labels as follows: (1) For i = 1, 2, · · · , k2 , φ∗ (v2i−1 v2i )=| φ(v2i−1 ) − φ(v2i ) | = N (2n − 4i) + 1, φ∗ (v2i−1 v2i−1 ) = (1) | φ(v2i−1 ) − φ(v2i−1 ) | = N (2n − 4i + 1) + (1) For i = 1, 2, · · · , k2 −1, φ∗ (v2i+1 v2i )=| φ(v2i+1 )−φ(v2i ) | = N (2n−4i−2)+1, φ∗ (v2i v2i ) = (1) | φ(v2i ) − φ(v2i ) | = N (2n − 4i − 1) + ∗ For i = k2 + 1, k2 + 2, · · · , n−1 , φ (v2i−1 v2i )=| φ(v2i−1 ) − φ(v2i ) | = N (2n − 4i + 1) + 1, (1) = | φ(v2i ) − φ(v2i ) | = N (2n − 4i) + (1) φ∗ (v2i v2i ) For i = k ∗ + 1, k2 + 2, · · · , n−1 , φ (v2i+1 v2i ) = | φ(v2i+1 ) − φ(v2i ) | = N (2n − 4i − 1) + For i = k ∗ + 1, k2 + 2, · · · , n+1 , φ (v2i−1 v2i−1 ) = | φ(v2i−1 ) − φ(v2i−1 ) | = N (2n − 4i + 2) + (1) (1) This show that the edges have the distinct labels {1, N + 1, 2N + 1, · · · , N (q − 1) + 1}, (1) where q = 2n − Hence for every positive integer N , Pn+ − vk is one modulo N graceful if n is odd and k is even (1) + Example 2.11 A one modulo graceful labelling of P13 − v2 is shown in Figure 134 V.Ramachandran and C.Sekar 70 67 61 15 55 21 49 27 43 33 37 64 12 58 18 52 24 46 30 40 36 Figure Case n is odd and k is odd Define   N (2i − 1) for i = 1, 2, · · · , k−1 φ(v2i ) =  N (k − 2) + N + 2N (i − ( (k+1) )) fori = k+1 k+3 , ,··· , n−1 , φ(v2i−1 ) = N (2n − 3) + − 2N (i − 1) for i = 1, 2, , n−1 , (1) φ(v2i−1 ) (1)   2N (i − 1) for i = 1, 2, · · · , k−1 =  2N ( k−1 − 1) + 3N + 2N (i − ( k+3 )) fori = 2 k+3 k+5 , ,··· , n−1 φ(v2i ) = 2N (n − 2) + − 2N (i − 1) for i = 1, 2, , n−1 , (1) The proof is similar to that of Case Hence for every positive integer N , Pn+ − vk is one modulo N graceful if n is odd and k is odd ¾ (1) + Example 2.12 A one modulo graceful labelling of P11 − v5 96 86 15 91 10 81 76 20 71 Figure 66 25 is shown in Figure 30 56 40 46 61 35 51 45 One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs 135 §3 Conclusion Subdivision or supersubdivision or arbitrary supersubdivision of certain graphs which are not graceful may be graceful The method adopted in making a graph one modulo N graceful will provide a new approach to have graceful labelling of graphs and it will be helpful to attack standard conjectures and unsolved open problems References [1] C Barrientos and S Barrientos, On graceful supersubdivisions of graphs, Bull Inst Combin Appl., 70 (2014) 77-85 [2] Joseph A Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics, 18 (2011), #DS6 [3] S.W.Golomb, How to number a graph in Graph theory and computing R.C Read, ed., Academic press, New York (1972)23-27 [4] R B Gnanajothi, Topics in Graph Theory, Ph.D Thesis, Madurai Kamaraj University, 1991 [5] G Sethuraman and P Selvaraju, Gracefulness of arbitrary supersubdivisions of graphs, Indian J Pure Appl Math., 32 (2001) 1059-1064 [6] G Sethuraman and P Selvaraju, Super-subdivisions of connected graphs are graceful, preprint [7] Z Liang, On the gracefulness of the graph Cm ∪ Pn , Ars Combin., 62(2002), 273-280 [8] V Ramachandran, C Sekar, One modulo N gracefullness of arbitrary supersubdivisions of graphs, International J Math Combin., Vol.2 (2014) 36-46 [9] V Ramachandran, C Sekar, One modulo N gracefulness of supersubdivision of ladder, Journal of Discrete Mathematical Sciences and Cryptography (Accepted) [10] C Sekar and V Ramachandren, Graceful labelling of arbitrary supersubdivision of disconnected graph, Ultra Scientist, 25(2)A (2013) 315-318 [11] C Sekar, Studies in Graph Theory, Ph.D Thesis, Madurai Kamaraj University, 2002 International J.Math Combin Vol.1(2015), 136-136 AMCA – An International Academy Has Been Established W.Barbara (Academy of Mathematical Combinatorics & Applications, Colorado, USA) The Academy of Mathematical Combinatorics & Applications (AMCA), initiated by mathematicians of USA, China and India in December of last year has been established in USA Its aim is promoting the progress and upholding the development of combinatorics and Smarandache multispaces with mathematical sciences, i.e., mathematical combinatorics, including algebra, topology, geometry, differential equations, theoretical physics, theoretical chemistry and mathematical theory on recycling economy or environmental sciences, for instance, the industrial ecology, and so as to advance a sustainable developing for global economy The activities of AMCA include organizing meetings of the membership, discussions on important topics, edit and publish scientific journals, special volumes and members monographs, awards and honours, recognizing 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Antidegree Equitable Sets in a Graph By C.ADIGA, K.N.S.KRISHNA 24 A New Approach to Natural Lift Curves of the Spherical Indicatrices of Timelike ă MUSTAFA C Bertrand Mate By MUSTAFA BILICI, EVREN ERGUN, ¸ ALIS¸KAN 35 Totally Umbilical Hemislant Submanifolds of Lorentzian (α)-Sasakian Manifold By B.LAHA, A.BHATTACHARYYA 49 On Translational Hull Of Completely J ∗, -Simple Semigroups By YIZHI CHEN, SIYAN LI, WEI CHEN 57 Some Minimal (r, 2, k)-Regular Graphs Containing a Given Graph and its Complement By N.R.SANTHI MAHESWARI, C.SEKAR 65 On Signed Graphs Whose Two Path Signed Graphs are Switching Equivalent to Their Jump Signed Graphs By P.S.K.REDDY, P.N.SAMANTA, K.S.PERMI 74 A Note on Prime and Sequential Labelings of Finite Graphs By MATHEW VARKEY T.K, SUNOJ B.S 80 The Forcing Vertex Monophonic Number of a Graph By P.TITUS, K.IYAPPAN 86 Skolem Difference Odd Mean Labeling of H-Graphs By P.SUGIRTHA, R.VASUKI, J.VENKATESWARI 96 Equitable Total Coloring of Some Graphs By GIRIJA G, V.VIVIK J 107 Some Characterizations for the Involute Curves in Dual Space ă IC I, MUSTAFA C By SULEYMAN S¸ENYURT, MUSTAFA BIL ¸ ALIS¸KAN 113 One Modulo N Gracefulness of Some Arbitrary Supersubdivision and Removal Graphs By V.RAMACHANDRAN, C.SEKAR 125 AMCA - An International Academy Has Been Established W.BARBARA 136 An International Journal on Mathematical Combinatorics .. .Vol. 1, 2 015 ISSN 19 37 -10 55 International Journal of Mathematical Combinatorics Edited By The Madis of Chinese Academy of Sciences and Academy of Mathematical Combinatorics & Applications... given by (1. 14) and using the fact that ≤ δ < 1, we obtain d(u1 , u2 ) = d(T u1 , T u2) ≤ eL d(u1 ,T u1 ) δ d(u1 , u2 ) + 2δ d(u1 , T u1 ) = eL d(u1 ,u1 ) δ d(u1 , u2 ) + 2δ d(u1 , u1 ) Fixed... ′′ The torsion is then given by τ 1 = det( 1 , 1 ′ , 1 ′′ ) , 1 ∧ 1 ′ τ 1 = √ 1 Ω2 where 1 = −2t1 κ λτ θ′ ′ W θ′ + W ′ W W θ′ + W θ′ κ + λτ W W θ′ + t1 κ λτ t2 κ + λτ ′ W θ′ 2 + W ′ W