Graceful valuations of 2-regular graphs with two components (Q1916092)

The authors prove that the theorem stated by A. Kotzig in 1984 and saying that the condition \(|E(G) |\equiv 0 \bmod 4\) or \(|E (G) |\equiv 3 \bmod 4\) is also sufficient for the gracefulness of a 2-regular graph \(G = (V(G), E(G))\) with exactly two components is true. Furthermore, the authors mention that the condition given above is not sufficient for 2-regular graphs with more than two components. This paper is concluded by proofs of some very interesting theorems dealing with graceful 2-regular graphs.