Gracefully Cultivating Trees on a Cycle

A graph G of size q is graceful if there exists an injective function f:V(G)→{0,1,…,q} such that each uv∈E(G) is labeled |f(u)−f(v)| and the resulting edge labels are distinct. Truszczyński conjectured that all unicyclic graphs except the cycle Cn, where , are graceful. In this paper, we present two methods to construct certain graceful unicyclic graphs when the length of cycles are congruent to 0 or 3 (mod 4).