Intern. J. Fuzzy Mathematical Archive Vol. 7, No. 1, 2015, 91-96 ISSN: 2320 –3242 (P), 2320 –3250 (online) Published on 22 January 2015 www.researchmathsci.org 91 International Journal of Gracefulness of Some Super Graphs of KC 4 -Snake A.Elumalai 1 and A.Anand Ephremnath 2 1 Department of Mathematics, Valliammai Engineering College Kattankulathur –603203, India 2 Department of Mathematics, Surya Group of Institutions School of Engineering and Technology, Vikiravandi, Villupuram – 605 652, India. Corresponding Author. e-mail: anand.ephrem@gmail.com Received 1 November 2014; accepted 4 December 2014 Abstract. In this paper we introduce a new definition called the complete m-points projection on some projected vertices of a graph and then we prove that the complete m- points projection on some projected vertices of 4 kC -snake is graceful. Keywords: Graphs, Complete m-points projection, n kC -snake AMS Mathematics Subject Classification (2010): 05C78 1. Introduction A function f is called a graceful labeling of a graph G with m edges if f is an injection from the vertex set of G to the set { } m , . . . , 2 , 1 , 0 such that, when each edge xyis assigned the label , ) ( ) ( y f x f - the resulting edge labels are distinct. Rosa [6] introduced such labeling in 1967 and named it as a β -valuation of graph while Golomb [5] independently introduced such labeling and called it as graceful labeling. Acharya [1] has constructed certain infinite families of graceful graphs from a given graceful graph while Rosa [7] and Golomb [5] have discussed gracefulness of complete bipartite graphs and Eulerian graphs. Sekar [8] has proved that the splitting graph (the graph obtained by duplicating the vertices of a given graph altogether) of C n admits graceful labeling for ) 4 (mod 2 , 1 ≡ n .A n kC -snakeis a connected graph with k blocks, each of the block is isomorphic to the cycle C n , such that the block-cut-vertex graph is a path. Following Chartrand, Lesniak [4], by a block-cut-vertex graph of a graph G we mean the graph whose vertices are the blocks and cut-vertices of G where two vertices are adjacent if and only if one vertex is a block and the other is a cut-vertex belonging to the block. We also call a n kC -snake as a cyclic snake. This graph was first introduced by Barrientos [3] and he proves that 4 kC -snakes are graceful and later it was discussed by Badr [2] as generalization of the concept of triangular snake introduced by Rosa [6]. A n kC -snake contains nk M = edges and 1 ) 1 ( - = k n N vertices. Among these vertices, 1 - k vertices have degree 4 and the other vertices of degree 2. Let 1 2 1 . . . , , - k u u u be the consecutive cut-vertices of G. Let d i be the distance between u i and