One modulo &ITN&IT gracefulness of Crowns, Armed crowns and Chain of even cycles

摘要

A function f is called a graceful labelling of a graph G with q edges if f is an injection from the vertices of G to the set {0, 1, 2, . . . , q} such that, when each edge xy is assigned the label vertical bar f (x) - f (y)vertical bar, the resulting edge labels are distinct. A graph G is said to be one modulo N graceful (where N is a positive integer) if there is a function phi from the vertex set of G to {0, 1, N, (N + 1) , 2N, (2N + 1), . . . , N(q - 1), N(q - 1) + 1} in such a way that (i) phi is 1 - 1 (ii) phi induces a bijection phi* from the edge set of G to {1, N + 1, 2N + 1, . . . , N(q - 1) + 1} where phi*(uv) = vertical bar phi(u) - phi(v)vertical bar. In this paper we prove that the crowns, armed crowns and chain of even cycles are one modulo N graceful for all positive integers N.