Some New Classes of Graceful Lobsters Obtained by Applying Inverse and Component Moving Transformations
C. V. Raman, West Midnapore, West Bengal · 2011
We observe that a lobster with diameter at least v e has a unique path H = x0x1 : : : xm with the property that besides the adjacencies in H both x0 and xm are adjacent to the centers of at least one K1;s, where s > 0, and each xi; 1 i m 1, is at most adjacent to the centers of some K1;s, where s 0. This unique path H is called the central path of the lobster. We call K1;s an even branch if s is nonzero even, an odd branch if s is odd, and a pendant branch if s = 0. In this paper we give graceful labelings to some new classes of lobsters with diameter at least v e, 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 lobsters appear in [5] also possess this property. However, in the lobsters of [5], at most the vertex x0 is attached to a combination of all three types of branches, whereas in this paper, we give graceful labelings to the lobsters in which not only the vertex x0 but also some (or all) xi; 1 i m, may exhibit this property.