Closed trail decompositions of some classes of regular graphs

Abstract

If H1,H2,…,Hk are edge-disjoint subgraphs of G such that E(G)=E(H1)∪E(H2)∪⋯∪E(Hk), then we say that H1,H2,…,HkdecomposeG. If each Hi≅H, then we say that H decomposes G and we denote it by H|G. If each Hi 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, (Km∘K¯n)(λ), into closed trails of lengths pm1,pm2,…,pmk, where p 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), 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 (Km∘K¯n)(λ), then it has a Tk-decomposition, where Tk 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 Tk-decomposition of (Km×Kn)(λ).