English

Abstract

Here we denote a diameter six tree by (a0; a1, a2, . . . , am; b1, b2, . . . , bn; c1, c2, . . . , cr) , where a0 is the center of the tree; ai, i = 1, 2, . . . ,m , bj, j = 1, 2, . . . , n , and ck, k = 1, 2, . . . , r are the vertices of the tree adjacent to a0 ; each ai is the center of a diameter four tree, each bj is the center of a star, and each ck is a pendant vertex. Here we give graceful labelings to some new classes of diameter six trees (a0; a1, a2, . . . , am; b1, b2, . . . , bn; c1, c2, . . . , cr) in which we find diameter four trees consisting of four different combinations of odd, even, and pendant branches with the total number of branches odd. Here by a branch we mean a star, i.e. 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.