(untitled) (Communications in Mathematical Research, 2021)
The present paper deals with the gracefulness of unconnected graph (jC_4n) ∪ P_m, and proves the following result: for positive integers n, j and m with n ≥ 1, j ≥ 2, the unconnected graph (jC_4n) ∪ P_m is a graceful graph for m = j − 1 or m ≥ n + j, where C_4n is a cycle with 4n vertexes, P_m is a path with m + 1 vertexes, and (jC_4n) ∪ P_m denotes the disjoint union of j − C_4n and P_m.