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 P with n vertices then for every vertex v∈V(P) and for every integer i∈{0,…,n-1} there is a graceful labelling of P such that v has label i. We show precisely when these labellings can also be α-labellings. We then extend this result to strong edge-magic labellings. In obtaining these results we make heavy use of π-representations of α-labellings and review some relevant results of Kotzig and Rosa.