On uniqueness of packing of three copies of 2-factors

The packing of three copies of a graph \(G\) is the union of three edge-disjoint copies (with the same vertex set) of \(G\). In this paper, we completely solve the problem of the uniqueness of packing of three copies of 2-regular graphs. In particular, we show that \(C_3,C_4,C_5,C_6\) and \(2C_3\) have no packing of three copies, \(C_7,C_8,C_3 \cup C_4, C_4 \cup C_4, C_3 \cup C_5\) and \(3C_3\) have unique packing, and any other collection of cycles has at least two distinct packings.