An L(p1, p2, p3, . . . , pm)- labeling of a graph G, has the vertices of G assigned with non-negative integers, such that the vertices at distance j should have at least pj as their label difference. If m = 3 and p1 = 3, p2 = 2, p3 = 1, it is called an L(3, 2, 1)-labeling which is widely studied in the literature. In this paper, we define an L(3, 2, 1)-path coloring of G as a labeling g : V (G) → Z+ such that between every pair of vertices there exists at least one path P where in the labeling restricted to this path is an L(3, 2, 1)-labeling. Among the labels assigned to any vertex of G under g, the maximum label is called the span of g. The L(3, 2, 1)-connection number of a graph G, denoted by k3c(G) is defined as the minimum value of span of g taken over all such labelings g. We call graphs with the special property that k3c(G) = |V (G)| as L(3, 2, 1)-path graceful. In this paper, we obtain k3c(G) of graphs that posses a Hamiltonian path and carry forward the discussion to certain classes of graphs which do not posses a Hamiltonian path, which is novel to this paper.