Let G := G(V, E) be an undirected finite simple graph with vertex set V and edge set E. Assume a function f from V into the set {0, 1,....,|E|} is injective, where |E| is the cardinality of E. If the set {|f(u)-f(v)|:uvE}={1,2,...,|E|}, then f is called graceful labeling for G, and the graph G is said to be graceful. A subset M of E such that any pair of elements of M are non-adjacent edges in E is called a 2matching in G. Furthermore, if every vertex of G is incident with exactly one element of M, then the matching M is said to be perfect. In this case, the graph G is called with perfect matching. Moreover, if a graceful graph G with graceful labeling f and with a perfect matching M has the property that for every uv ∈ M we have f(u)+f(v)=|E|, then G is called strongly graceful. In this talk we show that every graceful cycle with perfect matching is strongly graceful.

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