Graceful labellings of paths

It is easily shown that every path has a graceful labelling, however, in this paper we show that given almost any path PP with nn vertices then for every vertex v∈V(P)v∈V(P) and for every integer i∈{0,…,n-1}i∈{0,…,n-1} there is a graceful labelling of PP such that vv has label ii. We show precisely when these labellings can also be αα-labellings. We then extend this result to strong edge-magic labellings. In obtaining these results we make heavy use of ππ-representations of αα-labellings and review some relevant results of Kotzig and Rosa.