Closed trail decompositions of some classes of regular graphs

If H1,H2,…,HkH1,H2,…,Hk are edge-disjoint subgraphs of GG such that E(G)=E(H1)∪E(H2)∪⋯∪E(Hk)E(G)=E(H1)∪E(H2)∪⋯∪E(Hk), then we say that H1,H2,…,HkH1,H2,…,Hkdecompose  GG. If each Hi≅HHi≅H, then we say that HH decomposes GG and we denote it by H|GH|G. If each HiHi is a closed trail, then the decomposition is called a closed trail decomposition   of GG. In this paper, we consider the decomposition of a complete equipartite graph with multiplicity λλ, that is, (Km∘K¯n)(λ), into closed trails of lengths pm1,pm2,…,pmkpm1,pm2,…,pmk, where pp is an odd prime number or p=4,∑i=1kpmi is equal to the number of edges of the graph and ∘∘ denotes the wreath product of graphs. A similar result is also proved for (Km×Kn)(λ)(Km×Kn)(λ), where × denotes the tensor product of graphs, if there exists a pp-cycle decomposition of the graph. We obtain the following corollary: if k≥3k≥3 divides the number of edges of the even regular graph (Km∘K¯n)(λ), then it has a TkTk-decomposition, where TkTk denotes a closed trail of length kk. For m,n≥3m,n≥3, this corollary subsumes the main results of the papers [A. Burgess, M. Šajna, Closed trail decompositions of complete equipartite graphs, J. Combin. Des. 17 (2009) 374–403]; [B.R. Smith, Decomposing complete equipartite graphs into closed trails of length kk, Graphs Combin. 26 (2010) 133–140]. We have also partially obtained some results on TkTk-decomposition of (Km×Kn)(λ)(Km×Kn)(λ).