Chartrand et al. (2004) have given an upper bound for the nearly antipodal chromatic number ac′ (Pn) as (n−2 2 ) + 2 for n ≥ 9 and have found the exact value of ac′ (Pn) for n = 5, 6, 7, 8. Here we determine the exact values of ac′ (Pn) for n ≥ 8. They are 2p 2 − 6p + 8 for n = 2p and 2p 2 − 4p + 6 for n = 2p + 1. The exact value of the radio antipodal number ac(Pn) for the path Pn of order n has been determined by Khennoufa and Togni in 2005 as 2p 2 − 2p + 3 for n = 2p + 1 and 2p 2 − 4p + 5 for n = 2p. Although the value of ac(Pn) determined there is correct, we found a mistake in the proof of the lower bound when n = 2p (Theorem 6). However, we give an easy observation which proves this lower bound.
We observe that a lobster with diameter at least five has a unique path H = x0, x1, . . . , 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 adjacent at most to the centers of some K1,s, where s > 0. This 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 the existing literature only some specific classes of lobsters have been found to have graceful labelings. Lobsters to which we give graceful labelings in this paper share one common property with the graceful lobsters (in our earlier works) that each vertex xi , 0 ≤ i ≤ m − 1, is even, the degree of xm may be odd or even. However, we are able to attach any combination of all three types of branches to a vertex xi , 1 ≤ i ≤ m, with total number of branches even. Furthermore, in the lobsters here the vertices xi , 1 ≤ i ≤ m, on the central path are attached up to six different combinations of branches, which is at least one more than what we find in graceful lobsters in the earlier works.