A new class of graceful lobsters obtained from diameter four trees
Abstract
We observe that a lobster with diameter at least five has a unique path x0, x1, ⋯ ,xm (called the central path) such that x0 and xm are adjacent to the centers of at least one K1,s, > 0, and besides adjacencies in the central path each xi, 1 ≤ i ≤ m - 1, is at most adjacent to the centers of some K1,s, s ≥ 0. In this paper we give graceful labelings to some new classes of lobsters with diameter at least five, in which the degree of each vertex Xi, 0 ≤ i ≤ m - 1, on the central path is even and the degree of the vertex xm may be odd or even. The main idea used to obtain a graceful labeling of a lobster L here to form a diameter four tree T(L) from L by successively merging the vertices on the central path with x 0. give a graceful labeling to T(L) by arranging the stars incident on the center of T{L) in a proper order and using the techniques in [5], and finally get a graceful labeling of L by repeatedly applying component moving and inverse transformations.
