Some New Graceful Lobsters with Pendant Vertices with Central Paths

Abstract

The objective of this article is to give graceful labeling to some new classes of lobsters in a bid to resolve the three and half decade old Bermond’s conjecture that all lobsters are graceful. Here we use the method of component moving transformation such as the transfer of the first and second type and the derived transformations such as BD8TF, 1JTF, and 2JTF for generating graceful trees from a given one. In a bid to resolve Bermond’s conjecture here we give graceful labelings to many classes of lobsters possessing at least one of the two distinct features from those found in the literature as detailed below. 1.The central paths of the lobsters contain one or more vertices which do not have any neighbour apart from those on the central path. 2.One or more vertices of the central paths are attached to leaves. 3. The vertices on the central path may be attached to any of the fifteen different combination of odd, even, and pendant branches.