D-Graceful Labeling of a Path

[[abstract]]Let G be a undirected graph with a vertex set V and an edge set E. Given a nonnegative integer set D. A D-graceful labeling f of G is an injection f : V → D such that {|f(x) − f(y)| ¯¯ xy ∈ E} = {1, 2, 3, . . . , |E|}. A graph is called D-graceful if it has a D-graceful labeling. We call a graph G graceful if G is {0, 1, . . . , |E|}-graceful. Let Z(n; a, b) denote the set {0, 1, · · · , a−1, b+1, · · · , n+b−a}. In this paper, we showed that Pn is D-graceful for some D. And we conjecture that Pn is Z(n; t, t)- graceful except n = 2t = 2 and n = 3, 4