Some new graceful generalized classes of diameter six trees - Debdas Mishra; Sushant Kumar Rout; Puma Chandra Nayak (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2017)
Here we denote a diameter six tree by (c; a_1, a_2, …, a_m; b_1, b_2, …, b_n; c_1, c_2, …, c_r), where c is the center of the tree; a_i, i = 1, 2, …, m, b_j, j = 1, 2, …, n, and c_k, k = 1, 2, …, r are the vertices of the tree adjacent to c; 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. Here we give graceful labelings to some new classes of diameter six trees (c; a_1, a_2, …, a_m; b_1, b_2, …, b_n; c_1, c_2, …, c_r) in which a diameter four tree may contain any combination of branches with the total number of branches odd though with some conditions on the number of odd, even, and pendant branches. Here by a branch we mean a star, i.e. we call a star an odd branch if its center has an odd degree, an even branch if its center has an even degree, and a pendant branch if it is a pendant vertex.