Gracefulness of the Union of Cycles and Paths

Frucht and Salinas [1] conjectured that \(C(k) \cup P(n)\) (\(n \geq 3\)) is graceful if and only if \(k + n \geq 7\). We prove that \(C(2k) \cup P(n)\) is graceful for \(n > k + 1\) (\(k \geq 3\)).

For smaller cases we prove that \(C(2k) \cup P(n)\) is graceful for \(k = 3, 4, 5, 6; n \geq 2\).