Packing directed cycles of specified odd length into digraphs and alternating cycles into bipartite graphs

In this paper, we prove the following result. For given integers $k\ge 1,t\ge 0$ with $k\ge t$ and an odd integer $\ell \ge 3$, there exists an integer $n_0=n_0(k,t,\ell )$ satisfying the following: If D is a digraph of order $n\ge n_0$, and if $d_D^{+}\left(u\right)+d_D^{-}\left(v\right)\ge n+t$ for every two distinct vertices u and v with $(u,v)\notin A\left(D\right)$, then D contains k vertex-disjoint directed cycles of length ℓ or $\ell +1$ such that at least t of them are of length ℓ. This is a common extension of the results obtained by Brandt et al. (1997) and, Chiba and Yamashita (2018). We also discuss the relation between our result and problems on packing alternating cycles into bipartite graphs.