A class of diameter six trees exhibiting graceful labeling. (1st November 2016) - Panda, Amaresh Chandra; Mishra, Debdas; Dash, Rajani Ballav
Abstract: A diameter six tree may be represented by a set of vertices, say ( a 0 ; a 1, a 2, . . ., am; b 1, b 2, . . ., b n ; c 1, c 2, . . ., c r ), where a 0 stands for the center of the tree and the vertices a i, i = 1, 2, . . ., m, b j, j = 1, 2, . . ., n, c k, k = 1, 2, . . ., r are adjacent to a 0 . Moreover, each a i is the center of a diameter four tree, each b j is the center of a star, and each c k is a pendant vertex. Observe that in a diameter four tree, the neighbors of the center vertex are either the centers of some stars or pendant vertices. Here we call a star an odd branch if its center has an odd degree and an even branch if its center has an even degree. In the diameter six trees ( a 0 ; a 1, a 2, . . ., a m ; b 1, b 2, . . ., b n ; c 1, c 2, . . ., c r ) to which we give graceful labeling in this paper, each vertex a i, 1 ≤ i ≤ m, has an even degree and it may be adjacent to four different combinations consisting of odd branches, even branches, and pendant vertices.