Some new graceful generalized classes of diameter six trees

Abstract

Here we denote a {\it diameter six tree} by $(c; a_{1}, a_{2}, \ldots, a_{m};   b_{1}, b_{2}, \ldots,  b_{n}; c_{1}, c_{2}, \ldots, c_{r})$, where $c$ is the center of the tree; $a_{i}, i = 1, 2, \ldots, m$, $b_{j},  j =  1, 2, \ldots, n$, and $c_{k}, k = 1, 2, \ldots, r$ 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 $(c;  a_{1}, a_{2}, \ldots, a_{m};  b_{1}, b_{2}, \ldots, b_{n}; c_{1}, c_{2}, \ldots, c_{r})$ 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.