Some graceful lobsters with both odd and even degree vertices on the central path - Mishra, Debdas; Panigrahi, Pratima (Utilitas Mathematica, 2007)
We observe that a lobster with diameter at least five has a unique path H = χ0, χ1, ⋯ χm with the property that, besides the adjacencies in H, both χ0 and χm adjacent to the centers of at least one K1,S, where s > 0, and each χi 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 K1iS an even branch if s is nonzero even, an odd branch if s is odd, and a pendant branch if s = 0. The lobsters to which we give graceful labelings in this paper have one of the following features. Let l1 and l2 be the integers such that l1 ≤ l2 ≤ m. 1. The vertex χ0 Q is attached to an even number of odd branches and an odd number of pendant branches, the vertices χi, 1 ≤ i ≤ l1, are attached to any combination of odd and pendant branches (with some restriction). If l 1 < m, then one of the following holds. (i) The vertices χi, li + 1 ≤ i ≤ l2, are attached to any combination (with some restriction) of odd and even branches and each of the rest of the χi, if any, is attached to only odd (or even) branches, (ii) The vertices χi, l1 + 1 ≤ i ≤ l2, are attached to any combination of even and pendant branches (with some restriction) and each of the rest of the χi, if any, is attached to only even branches. 2. The vertex χ0 is attached to a combination of odd, even, and pendant branches, and combinations of branches incident on the vertices χi, 1 ≤ i ≤ m, are the particular cases of those in (1).