Closed trail decompositions of some classes of regular graphs

If H 1 , H 2 , … , H k are edge-disjoint subgraphs of G such that E ( G ) = E ( H 1 ) ∪ E ( H 2 ) ∪ ⋯ ∪ E ( H k ) , then we say that H 1 , H 2 , … , H k decompose G . If each H i ≅ H , then we say that H decomposes G and we denote it by H | G . If each H i is a closed trail, then the decomposition is called a closed trail decomposition of G . In this paper, we consider the decomposition of a complete equipartite graph with multiplicity λ , that is, ( K m ∘ K ¯ n ) ( λ ) , into closed trails of lengths p m 1 , p m 2 , … , p m k , where p is an odd prime number or p = 4 , ∑ i = 1 k p m i is equal to the number of edges of the graph and ∘ denotes the wreath product of graphs. A similar result is also proved for ( K m × K n ) ( λ ) , where × denotes the tensor product of graphs, if there exists a p -cycle decomposition of the graph. We obtain the following corollary: if k ≥ 3 divides the number of edges of the even regular graph ( K m ∘ K ¯ n ) ( λ ) , then it has a T k -decomposition, where T k denotes a closed trail of length k . For m , 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 k , Graphs Combin. 26 (2010) 133–140]. We have also partially obtained some results on T k -decomposition of ( K m × K n ) ( λ ) .