Decomposition of complete bipartite graphs into paths and cycles

We give conditions for decomposability of the complete bipartite graph Km,nKm,n into paths and cycles having kk edges. In particular, we find necessary and sufficient conditions for such decomposition in Km,nKm,n, when m≥k2,n≥⌈k+12⌉ for k≡0(mod4) and when m,n≥2km,n≥2k for k≡2(mod4). As a consequence, we show that for nonnegative integers pp and qq, an even integer kk, and odd nn with n>4kn>4k, there exists a decomposition of the complete graph KnKn into pp paths and qq cycles both having kk edges if and only if k(p+q)=n2 and p≠1p≠1.